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THE 


Elements  of  Algebra 


BY 


GEORGE   LILLEY,  Ph.D.,  LL.D. 


EX-PRESIDENT    SOUTH     DAKOTA     AGRICULTURAL    COLLBOB 


OF  THE     ^ 

UNIVERSITY 


SILVER,   BURDETT    &    COMPANY 
New  York  .  .  .  BOSTON  .  .  .  Chicago 
1894 


Copyright,  1892, 
By  Silver,  Burdett  and  Company. 


mniijcrsitg  lirrss : 
John  Wilson  and  Son,  Cambridqk,  U.S.A. 


PREFACE. 


Algebra  is  a  means  to  be  used  in  other  mathematical 
work;  it  develops  the  mathematical  language,  and  is  the 
great  mathematical  instrument.  If  the  student  would  be- 
come a  mathematician,  he  must  understand  this  language 
and  possess  facility  in  handling  the  various  forms  of  literal 
expressions. 

Attention  is  called  to  the  sequence  of  subjects  as  herein 
presented.  Involution  is  introduced  as  an  application  of 
multiplication,  evolution  as  an  application  of  division,  and 
logarithms  as  an  application  of  exponents.  Throughout  the 
book  the  student  is  led  to  see  that  one  subject  follows  as  an 
application  of  another  subject.  The  beginner  is  led  to  see  at 
the  outset  that  Algebra,  like  Arithmetic,  treats  of  numbers. 

Algebraic  terms  and  definitions  are  not  introduced  until 
the  student  is  required  to  put  them  into  actual  use.  Correct 
processes  are  cleariy  set  forth  by  carefully  prepared  solutions, 
the  study  of  which  leads  the  pupil  to  discover  that  method 
and  theory  follow  directly  from  practice,  and  that  methods 
are  merely  clear,  definite,  linguistic  descriptions  of  correct 
processes. 

The  book  is  sufficiently  advanced  for  the  best  High  Schools 
and  Academies,  and  covers  sufficient  ground  for  admission  to 
any  American  College. 

Great  care  has  been  given  to  the  selection  and  arrangement 
of  numerous  examples  and  problems.     These  have  been,  for 

18:5963 


iv  PREFACE. 

the  most  part,  tested  in  the  recitation-room,  and  are  not  so 
difficult  as  to  discourage  the  beginner. 

It  remains  for  the  author  to  express  his  sincere  thanks 
to  W.  H.  Hatch,  Superintendent  of  Schools,  Moline,  111. ;  to 
Professor  W.  C.  Bojden,  Sub-Master  of  the  Boston  Normal 
School,  Boston,  Mass. ;  and  to  O.  S.  Cook,  connected  with 
the  literary  department  of  Messrs.  Silver,  Burdett  &  Co.,  for 
reading  the  manuscript  and  for  valuable  suggestions. 

GEORGE   LILLEY. 

Pullman,  Washington,  June,  1892. 


PREFACE  TO  THE   SECOND  EDITION. 

In  this  edition  the  typographical  errors  have  been  cor- 
rected, and  a  page  of  examples  has  been  added  to  Chapter 
XXVII ;  also,  the  exercises  have  been  carefully  revised  and 
corrected.  Answers  to  the  examples  and  problems  have  been 
prepared,  and  are  bound  in  the  book,  or  separately  in  flexi- 
ble cloth  covers.  The  answer-book  is  furnished  for  the  use 
of  the  class  only  on  application  of  teachers  to  the  publishers 
for  it.  The  publishers  and  the  author  desire  to  express  their 
appreciation  of  the  very  favorable  reception  accorded  to  the 
first  edition. 

September,  1894. 


CONTENTS. 


CHArrEK  PACK 

I.    First  Principles l 

II.    Algebraic  Addition 19 

III.  Algebraic  Subtraction 27 

IV.  Algebraic  Multiplication 35 

V.    Involution 52 

VI.    Algebraic  Division 60 

VII.    Evolution 79 

VIII.    Use  of  Algebraic  Symbols 99 

IX.    Simple  Equations 104 

X.    Problems  Leading  to  Simple  Equations 109 

XI.    Factoring 119 

XII.    Highest  Common  Factor 141 

XIII.  Lowest  Common  Multiple 155 

XIV.  Algebraic  Fractions '164 

XV.    Fractional  Equations •.    .    .  201 

XVI.    Simultaneous  Simple  Equations 215 

XVII.  Problems  Leading  to  Simultaneous  Equations     .    .  238 

XVIIL    Exponents 248 

XIX.    Radical  Expressions       263 

XX.    Logarithms 296 

XXI.    Quadratic  Equations 312 


vi  CONTENTS. 

CHAPTEK  PAGE 

XXII.    Equations  which  may  be  Solved  as  Quadratics      .  330 

Theory  of  Quadratic  Equations 339 

XXIII.  Simultaneous  Quadratic  Equations 345 

XXIV.  Indeterminate  Equations 355 

XXV,    Inequalities 363 

XXVI.    Series 373 

Arithmetical. 373 

Geometrical 379 

Harmonical 384 

XXVII.    Ratio  and  Proportion 388 


APPENDIX 401 


INDEX  TO   DEFINITIONS. 


PAOI 

Algebra 118 

Binomial 46 

Coefficient 20 

Equation,  Biquadratic       .          334 

"  Degree  of,  Roots  of 35,  108 

"          Exponential 309 

Literal 206 

"         Syminetriciil 347 

Expression,  Algebraic 90 

"            Compound 23 

**            Homogeneous 349 

**            Imaginary 286 

**            Irrational 263 

**            Mixed 164 

"            Simple 21 

Factor 119 

Figures,  Subscript 227 

Fraction,  Complex 188 

Continued 190 

Identities 104 

Index 79 

Mean,  Arithmetical 378 

"      Geometrical 383 

"      Harmonical 385 

Nlonoraial 21 

Multiple ,155 

"        Common 155 

Multiplication,  Algebraic 38 


Vlll  INDEX   TO   DEFINITIONS. 

PAGE 

Numbers,  Algebraic,  Absolute ...  14 

"          Known 107 

"          Negative 11 

"          Scale  of 12 

"          Unknown 107 

Polynomial 23 

Power 35 

Progression,  Arithmetical 373 

"            Geometrical 379 

"             Harmonical 384 

Quantity <.....• 389 

Keciprocal ^8 

Roots 79,  340 

Signs,  Algebraic 13 

"       Double 80 

"       Law  of .     38,  61 

"       Radical 79 

Subtraction ' 34 

Surd,  Similar 263 

"     Entire,  Mixed 264 

"     Quadratic 290 

Symbols  of  Abbreviation 7 

"        of  Aggregation,  of  Relation 6 

"         of  Operation 1,  99 

Terms 3,  90 

"       Like 20 

Term,  Absolute 207 

"       Degree  of.  Dimension  of 345 

Value,  Absolute •     <  14 

**      Numerical 9 


ELEMENTS  OF  ALGEBRA 


CHAPTER   I. 
FIRST    PRINCIPLES. 

1.  In  Algebra  figures  and  letters  are  used  to  represent 
numbers,  instead  of  figures,  as  in  Arithmetic. 

Thus,  we  may  use  x  to  represent  the  number  of  dollars  in  a  man's 
business,  the  number  of  cents  in  the  cost  of  an  article,  the  number  of 
miles  from  one  place  to  another,  the  number  of  persons  in  our  class, 
etc. 

In  Algebra,  the  letter  x  is  reasoned  about  and  operated  upon  just 
the  same  aa  the  numbers  which  it  represents  are  reasoned  about  and 
operated  upon  in  Arithmetic. 

2.  Symbols  of  Operation.  The  signs  +,  — ,  X,  and  -h, 
are  used  to  deuute  the  algebraic  operations  addition,  sulv 
traction,  multiplication,  and  division,  that  in  Arithmetic 
can  actually  be  performed.  +  is  read  'plus;  —  is  read 
miniis ;  X  is  read  multiplied  by;  H-  is  read  divided  by. 
A  dot  or  point  is  sometimes  used  instead  of  the  sign  X. 
Thus,  a  X  6  and  a  •  h  both  mean  that  a  is  to  be  multiplied 
by  h.  The  multiplicand  is  usually  written  before  the 
multiplier. 

Dimsio-ii  in  Algebra  is  more  frequently  represented  by 
placing  the  dividend  as  the  numerator,  and  the  divisor  as 


2  ELEMENTS   OF  ALGEBRA. 

the  denominator  of  a  fraction.     Thus,  a  -i-  b,  or  - ,  means 

b 

that  a  is  to  be  divided  by  b.     Eead  a  divided  by  b. 

m 
Note.     Do  not  read  such  expressions  as  —  ,  m  over  n;  it  is  meaningless. 

3.  We  must  be  careful  to  distinguish  between  arith- 
metical and  algebraic  operations.  The  former  can  actually 
be  performed,  whereas  many  operations  in  Algebra  can  only 
be  indicated. 

Thus,  suppose  a  man  owes  ^  5  for  a  vest  and  %  20  for  a  coat,  actual 
addition  gives  $25  as  his  total  indebtedness.  But  if  the  number  of 
dollars  he  owes  for  the  vest  be  represented  by  m,  and  the  number  of 
dollars  that  he  owes  for  the  coat  be  represented  by  n,  his  entire  debt 
can  only  be  indicated.  In  order  to  show  that  the  number  represented 
by  m  is  to  be  added  to  the  number  represented  by  n,  we  use  the  sign 
+  written  between  them ;  thus,  m  +  n. 


Exercise  I. 

Eead  the  following  algebraic  expressions  : 

1.  0!  +  100  ;  a  +  10  -  2  ;   &  -  2  ;  &  -  100  +  8. 

2.  a  -\-b;   m  -\-  n  -{-  ^  \    m  +  s  ~  r;    a  —  b  +  m. 

3.  c  +  2x5;   c -10  X2;  s-nX  r-20. 

4.  q  +  t  +  S  X  m;  c  + m^n  —  s  -  q;  —  -  +  c-^a—p -{- 1  -  x. 

a 

Indicate  by  means  of  algebraic  expressions  the  following: 

5.  The  sum  of  m  and  n.  The  difference  between  m 
and  n.     The  sum  of  x,  y,  and  a. 

6.  The  sum  of  m,  n,  and  r  diminished  by  t.  If  you  had 
m  cents,  earned  n  cents,  and  are  given  r  cents,  and  then 
spend  t  cents ;  how  many  cents  will  you  have  left  ? 


FIRST   FRINCIPLKS.  3 

7.  John  has  m  apples,  Henry  has  n  apples,  and  Charles 
has  b  apples ;  express  the  number  of  their  apples.  How 
many  more  have  John  and  Henry  than  Charles  ? 

8.  If  you  buy  goods  for  a  dollars  and  sell  them  at  a  gain 
of  b  dollars,  express  the  selling  price. 

9.  I  buy  goods  for  m  dollars  and  sell  them  at  a  loss  of 
71  dollars ;  express  my  selling  price. 

10.  Henry  had  x  marbles ;  he  gave  John  vi  marbles,  and 
Charles  ii  marbles.     How  many  had  he  left  ? 

11.  I  pay  n  cents  for  a  reader,  x  cents  for  a  history,  y 
cents  for  a  grammar,  6  cents  for  car-fare,  and  have  m  cents 
left ;  express  the  number  of  cents  that  I  had  at  first. 

12.  A  boy  earned  a  dollars,  then  received  m  dollars 
from  his  father,  n  dollars  from  his  mother ;  and  spent  k 
dollars  of  what  he  had  for  books,  x  dollars  for  a  coat,  and 
y  dollars  for  a  sled.  Express  the  number  of  dollars  he  had 
left. 

4.  The  Sign  of  Multiplication  is  generally  omitted  in 
Algebra,  except  between   figures.     Thus, 

bah  means  bXaXb,  prstuz  means  p  X  r  X  s  X  t  X  v  X  z  ; 
J  •  3  •  4  •  5  means  2  X  3  X  4  X  5,  or  120. 

Again,  if  the  numher  of  gallons  in  a  cisk  of  cider  is  represented  by 
a,  and  the  number  of  cents  in  the  cost  of  one  gallon  is  represented 
by  m,  then  the  number  of  cents  in  the  cost  of  n  casks  is  represented 
by  amn. 

5.  In  the  expression  5  +  2«o  —  a+ -— —  :  5, 

m  2  am  ^ 

2  ab,  a,  — ,  and  -ttt-  &re  called  Terms. 
n  ooc 


ELEMENTS   OF  ALGEBRA. 


Exercise  2. 

Eead  and  state  the  meaning  of  the  following  algebraic 
expressions : 

1.  5ahx-\ ah.    Result :  5  times  a  times  h  times 

c 

X,  plus  7n  times  n  divided  by  c,  minus  a  times  b ;   etc. 

2.  kl  + 1-t;  PQrs  + ab cd  +  mnx y  —  80. 

en       b 

,      ^  ^  ^klx    abed     a.  i  .  i-i 

3.  amnpgr  —  cdXo-\-- ; Zb d+ 11— r. 

,  bwyz    mnop 

4    bx'^+12pqrst-Q^hk-^a-\-imz. 
ab 

„    S  ab  d  —  10  mnr  +  Imnr  st 

5.   . 

ad  —  1 

6.  h  +  '^-^+u;  2^+2^^+^^.  ?±fZ^*  +  „  +  *_±f . 

4  y  u  X  a  I 

6.  It  is  customary  to  write  the  letters  in  the  order  of 
the  alphabet. 

In  a  product  represented  by  several  letters  and  numbers,  the  num- 
bers are  written  first.     Thus, 

cX&XaX5X3  is  written  3  X  5  ahc  ;  both  mean  15  a  6  c. 
Also,  s  X  r  X  n  X  m  X  25  is  written  25  mnr s. 


Exercise  3. 

Write  algebraic  expressions  for  the  following : 

1.  The  product  of  x,  y,  and  z ;  of  m,  n,  and  5 ;  of  3  and 
xy,  of  5,  a,  b,  and  S  X  mn, 

2.  The  product  of  a  and  b  divided  by  their  sum.     Their 
product  divided  by  their  difference. 


FIRST  PHINCIPLES.  5 

3.  The  product  of  m,  n,  r,  and  25  divided  by  the  sura  of 
m  and  n.  The  same  product  divided  by  the  difference  of 
VI  and  71. 

4.  A  travels  at  the  rate  of  3  miles  an  hour ;  how  many 
hours  will  it  take  him  to  travel  30  miles  ?  How  many 
hours  to  travel  a  miles  ?  To  travel  m  n  miles  ?  To  travel 
60  a  in  n  miles  ? 

5.  A  man  bought  18  loads  of  wheat,  of  m  bushels  each, 
at  n  cents  a  bushel ;  how  many  cents  in  the  entire  cost  ? 

6.  In  example  5,  suppose  that  he  sold  the  wheat  at  a 
gain  of  r  cents  a  bushel ;  how  many  cents  did  he  gain  ? 
How  many  cents  in  the  selling  price  ? 

7.  In  example  5,  suppose  that  he  sold  the  wheat  at  a 
loss  of  a  cents  a  bushel ;  how  many  cents  would  he  lose  ? 
how  many  cents  in  the  selling  price  ? 

8.  A  man  bought  a  boxes  of  peaches,  each  containing  h 
peaches,  at  c  cents  a  peach ;  and  m  baskets  of  grapes,  each 
containing  n  pounds,  at  r  cents  a  pound.  How  many  cents 
did  he  pay  for  both  ? 

9.  A  man  worked  n  hours  a  day  for  m  days,  at  a  cents 
an  hour.  With  the  money  he  bought  a  coat  for  x  cents  ; 
how  many  cents  had  he  left? 

10.  One  boy  sold  a  apples  at  c  cents  each  ;  another  sold 
n  peaches  at  m  cents  each  ;  a  third  sold  r  peai-s  at  t  cents 
each.     How  many  cents  did  they  all  receive  ? 

11.  I  buy  5  tons  of  coal  at  SIO  per  ton,  and  pay  for 
it  in  cloth  at  S2  per  yard  ;  how  many  yards  will  it  take? 
I  buy  a  tons  of  coal  at  h  dollars  per  ton,  and  pay  for  it  in 
cloth  at  m  dollars  a  yard ;  how  many  yards  will  it  take  ? 


6  ELEMENTS  OF  ALGEBRA. 

12.  A  man  works  n  weeks  at  h  dollars  a  week,  and  his 
son  works  m  weeks  at  r  dollars  a  week.  With  the  money 
they  pay  for  c  cords  of  wood  at  d  dollars  a  cord ;  how  many 
dollars  have  they  left  ? 

13.  If  5  cords  of  wood  cost  $15,  how  many  dollars  will 
3  cords  cost?  If  c  cords  cost  $m,  how  many  dollars  will 
n  cords  cost  ? 

14.  A  man  drove  3  hours  at  the  rate  of  10  miles  an 
hour ;  how  many  hours  will  it  take  him  to  walk  back  at 
the  rate  of  6  miles  an  hour  ?  If  he  drives  3  days  n  hours 
each  day,  at  the  rate  of  t  miles  an  liour,  and  5  days  m  hours 
each  day,  at  the  rate  of  s  miles  an  hour,  how  many  hours 
will  it  take  him  to  return  over  the  same  distance,  at  the 
rate  of  r  miles  an  hour  ? 

15.  If  you  buy  t  tons  of  coal  at  the  rate  oi  %d  for  n 
tons,  and  sell  it  at  a  loss  of  $  Z  on  each  ton,  how  many 
dollars  will  you  receive  ?  Suppose  you  sell  at  a  gain  of 
%h  on  each  ton,  how  many  dollars  will  you  get  for  it  ? 
Suppose  you  sell  all  of  it  for  r  dollars,  and  make  a  profit, 
how  many  dollars  profit  will  you  get  ? 

7.    Symbols  of  Relation.     The  signs   =,  >,  and  <,  are 

used  for  the  words,  equals,  is  greater  than,  and  is  less  than, 
respectively. 

Symbols  of  Aggregation.  The  signs  (  ) ,  [  ] ,  {  } ,  and  , 
are  used  to  show  that  the  terms  enclosed  by  them  are  to 
be  treated  ns  one  number.  They  are  called  parenthesis, 
bracket,  hracc,  and  vinculum,  respectively.     Thus, 

(2  ft  +  6)  (3  X  -  y),  [2  a  +  h][3x  -  yl  {2  a  +  b]  {3  x  ~  i/], 
2a  i-  b  X  3x  —  y,  each  shows  that  the  number  obtained  })y  adding 
the  terms  2  a  and  b  is  to  be  multiplied  by  the  result  obtained  by 
subtracting  y  from  3  x. 


FIRST   PRINCIPLES.  7 

Sjrmbols  of  Abbreviation.  The  signs  (of  deduction)  .*., 
(of  reason)  •.*,  and  (of  continuation)  ....,  are  used  for  the 
words,  heiice  or  therefore,  since  or  hecmise,  and  so  on, 
respectively. 

8.  Since  81  =  9  X  9,  or  written  9^  for  brevity,  81  is 
called  the  second  power  of  9.  Since  27  =  3  X  3  X  3,  or 
written  3^  for  brevity,  27  is  called  the  third  power  of  3. 
Similarly  a^,  {m  n)\  (m  +  n)^,  are  called  second  poioers  of 
a,  m  n,  and  m  -^  n\  also  a^,  (m  n)^,  {m  -\-  n)^,  are  third 
powers  of  a,  m  n,  and  m  -\-  n.  cfi  means  a  X  a  ;  a^  means 
a  X  a  X  a  \  etc.  In  general,  ft"  is  called  the  nth  power  of 
a,  read  a  7ith  power. 

9.  In  the  expression  a^  +  h^c^  —  3  x";  2,  4,  5,  and  ?i 
are  called  Exponents,  h^ c^  means  bxbxbxbxcxc 
xcX  cXc;  4  and  5  are  used  for  convenience  to  show  how 
many  times  b  and  c  are  used  as  factors. 

We  must  be  careful  to  keep  in  mind  the  meaning  of  each  indicated 
operation  when  rea(hng  an  algebraic  expression.  Thus,  the  expres- 
sion 5  X*  j/"^  —  2  a* 6  («'  —  6*)^  +  3a^c*d"*  means,  five  times  the  third 
power  of  X  times,  the  second  power  of  y^  minus  two  tim^s  the  fourth 
power  of  a  times  b  times  the  fifth  power  of  tlie  expression  in  the 
parenthesis,  a  seventh  power  minus  b  sixth  power,  phis  three  times 
the  fifth  power  of  a  times  the  fourth  power  of  c  times  the  with  power 
of  d. 

Exercise  4. 

Read  the  following : 

1.  m^;3m^x^;5m^nY,'fh(M^(^iab^-hh,vL^-n^\  lOaVr^. 

2.  m^n^  -f  5  a^bxi/  -  3  m^^  x^;   m^n^  -'lab  m  n  +  (H'K 

3.  10  (ft  6)^0;  (?n3n8)  (m  nf\  (a^  -  „)2.  (,„2  _  3  ^^^a 

4.  (m  n  -  m3)3;  3  (v^  b  {<i  -  }?'^-  (n^  +  />")  (a^  -  b^f- 


8  ELEMENTS  OF  ALGEBRA. 

5.  3  oTlf',  3  {a  bf ;  a^  (b^  -  c^  -  d'^f^  (m^  n^)  (m  n)\ 

6.  (10  m  +  n"^)  (10  n^  -  m^f  <  15  a  (x  -  i/)^  (x  +  yf  ; 

(1^  +  1)^  +  d^  +  ^7         (c5  +  d^y 

b{a  +  h  +  ef     ^  ^''  ^     c2  +  6^3    ' 

7.  •.•a  +  2:c=:6  +  ^,  .'.x  =  h  —  a\  {a'^  —  c''f={7n?+n^f, 

...  a"*  -  C'*  =r  m2  ^  ^2  .     ^2  -  ^-3  =  2  ^3  _  2  ,^,2^  .-.0.^3  =  ,^,^2  . 

x-\-x-^x-\'X-\- to  n  terms  =^  nx\    a  X  a  X  a  X  a  X 

....  to  71  factoi'S  =  a" ;    1  -^  x  ■\-  x^  +  st?  -\-  .... 


..2    I    ^,  ^^    I  f^  .,  f^,„«o  _  ^('''"  ~  ^) 


1  -a? 

a  +  ar  +  a  ?'^  +  a ?^  +  . . . .  to  7«.  terms  =  ,     . 

r  —  1 

Write  algebraic  expressions  for  the  following : 

8.  The  sum  of  m  and  n.  The  double  of  x.  The  second 
power  of  the  sum  of  a  and  h.  The  second  power  of  differ- 
ence between  x  and  y.  Five  times  the  third  power  of  the 
difference  of  x  and  y. 

9.  The  second  power  of  the  sum  of  x  second  power 
and  y.  The  second  power  of  the  sum  of  x  and  y  second 
power.  The  product  of  the  fourth  power  of  x,  the  third 
power  of  ?/,  and  the  second  power  of  7n.  The  product  of 
the  first  power  of  x  and  three  times  the  nth  power  of  y. 
The  product  of  x  second  power  plus  y  second  power,  and 
X  second  power  minus  y  second  power. 

10.  The  product  of  the  sum  of  x  second  power  and  y 
by  n  a.  Five  n  third  power  minus  seven  m  n  plus  six  a 
second  power,  m  third  power  minus  two  times  b  second 
power  c  plus  n  fourth  power  is  equal  to  n  times  y. 

11.  Seven  times  m  fourth  power  times  n  second  power 
minus  two  times  -^t  seventh  power  times  m  third  power 
plus  three  times  a  tliird  power  times  h  second  power  plus 
eight  times  a  second  power  times  b  third  power  plus  five 


FIRST  PRINCIPLES.  9 

times  a  fifth  power.  Since  a  plus  h  equals  m  minus  w, 
therefore  the  second  power  of  a  plus  h  is  equal  to  the 
second  power  of  m  minus  ii. 

12.  Therefore,  x  is  equal  to  m  third  power,  because  x 
plus  three  m  third  power  is  equal  to  two  x  plus  two  m 
third  power,  a  plus  a  plus  a,  and  so  on  to  n  minus  two 
terms,  equals  n  minus  two  times  a.  The  second  power 
of  m  plus  yi,  divided  by  m  minus  n  is  less  or  greater  than 
m  times  a  plus  h  plus  c  plus  o?  plus  e.  a  less  than  6  is 
equal  to  iii  greater  than  n. 

13.  A  horse  eats  a  bushels  and  an  ox  h  bushels  of  oats 
in  a  week  ;  how  many  bushels  will  they  together  eat  in  n 
weeks  ?  If  a  man  was  a  years  old  50  years  ago,  how  old 
will  he  be  x  years  hence  ? 

10.  The  Nomerical  Value  of  an  algebraic  expression  is 
the  number  of  positive  or  negative  units  it  contains,  and  is 
found  by  giving  a  particular  value  to  each  letter,  and  then 
peiforming  the  operations  indicated.     Thus, 

If  a  -  3,  6  =  4,  x  =  5,  y  =  6,  find  the  numerical  values  of  : 

,2,.       9  6x« 
25  a*  1/2 

Replacing?  the  letters  in  each  expression  hy  the  •particular  values 
given  for  them,  we  have 

Process.  4  a«6«  =  4  x  3^  X  4» 


=  4X9X64 
=  2304. 


9h3*         9  X  4  X  5« 
25rt»y2~25  X  3»X  62 
9  X  4  X  125 


25  X  27  X  36 

=  A 
27' 

U.   If  one  factor  of  a  product  is  equal  to  0,  the  whole 

product  nmst  be  equal  to  0,  wlmtever  values  the  other  factors 


10  ELEMENTS   OF  ALGEBRA.      , 

may  have ;  and  it  is  also  clear  that  no  product  can  be  zero 
unless  one  of  the  factors  is  zero.  Thus,  ah  is  zero  if  a  is 
zero,  or  if  b  is  zero;  and  if  ab  is  zero,  either  a  or  6  is  zero. 
Again,  if  a;  =  0,  then  a^h'^xy^  =  0,  also  ax{y^  +  62;  +  «^) 
==  0,  whatever  be  the  values  of  a,  7;,  3/,  and  z. 

Exercise  5. 

If  <x  =  6,  &  =  2,  c  =  1,  a;  =  5,  ?/  —  4,  find  the  numerical 
values  of  the  following  algebraic  expressions  : 

L    3c2;  72/3;  5a&;  9^?/,  8Z?3.  3-^5.  ^^8.7^4.  ^^^.10.  3  ^^4^ 

2.  9  6*;  2  a  a; ;  3/^  ,  10  x^  ;  \y^\  5  6?/;  |^  r^^  ;  ^^ab  cxy. 

3.  3^2  ^^ 5  1^  ^  ^ ;  7  c^  ;  f  .«3  ;  a*  x^  ;  8  a^^  2  3/*  ;  |  acxy. 

li  7n  =  2,  n  =  3,  p  =  I,  q  =  0,  r  =  4,  s  =  6,  find  the 
values  of: 

4    ^^.  Pi,„,2^.  4^^^■^  ?ZL!!_^  ^J!^.   o«»9n.   2m2g 

p;     8^'^^      r,        6     06     5       s    4    5  7776  r-     27  m'-    64     2' 
9  7?t3 '  6  '  '  /^       '     54  ^m  '      32    '   r»  '  3" 

Example  6.  Find  the  value  of  5  6^  +  j%  x  y  -  5  a^  -  ^a^b^, 
when  (1  =  2,   6  =  3,  x  =  5,  and   i/  =  10. 

Replacing  the  letters  in  the  expression  by  the  particular  values 
given  for  them,  we  have 

Process. 

563  4-  ^3_xy  _ 5a2_ I Q^'2^3  =  5  X  33  +^3_  X  5  X  10  -  5  X  22  -  I  X  22  X  33 
=  5X27  +  3X5  -5X4-3x27 

=  49. 

Example  7.  Find  the  value  of  mny^-\- mhi xyt  +  m"nh's  —  -j-^ , 
wlien  ?7i  =  5,  u  =  2,  r  =  3,  s  =  4,  X  =  0,  and  y  =  I. 


FIRST   PRINCIPLES.  11 

Process. 

mny*-Hm*nxytfm%V8-^  =  5  X  2  X  IHO  +  S-'X  2»  X  3<X4-^  ^  ^.^ 

25  X« 
=  5X2X1  +  25X8X81X4-——^ 

=  10  4-  64800  -  50 
=  64760. 

If  a  =  1,  b  =  2,  c  =  3,  and  d  =  0,  find  the  numerical 
values  of  the  following  algebraic  expressions  : 

8.  10  a  —  4:b -{- 6  c -\- 5  d]   ab  +  hc  +  ac  —  da, 

9.  6  ab-Scd-{- 10  a  d-2b  c+  2bd;  2  be  +  10  cd. 
10.    a^  +  Ir^-\-c^-(P;  abc  +  10  bed  +  5  ac d  + 'S  abd. 

.11.    a*  +  h^  -\-  c'-d;  +ob-8e  -{-  ad;  —40ad-\-ab. 

12.  5a+3c-G6+6(/;   36co?-|-2acc?-10rt&rf. 

13.  5/>r3  +  .^3  ^_  /,3_  i.25a&3c;  15  «2  ^_  jj.  r*  +  10  a  &. 

.Cw/2/,4  .IftS 


14.    c3-  8  «(/6^- 5  ftio^.  j^^y. 


c3  ai«  62  c  ' 


15.    125a6cc/*m  +  ^,-^^l^';  «»  4.  ?,3  +  ,3  +  ,^;^. 
80  X 

V\    2  .,  ?,3  4.   iL  ^  2;{c  -  ^^,   3  «2i3,;6^  ^  '_'  _  2^' 


/;  a' 


8      ,      ,       3 


)  a 


**  2^-'^  c^      abc 

8  ^/  /•"    8<*       ^'^       r^ 

19.     J  a  f^  ;/  -\-  I  n  —  |  ^^2  d^xy\    Wa^^^^  '^  ~q~  ~  ^- 

o  o 

12.    Negative  Numbers,     if  a  person  owes  a  debt  of  ten  dollars, 
and  \\di»  but  .six  dollars  in  money,  he  can  pay  the  debt  only  in  part. 


12 


ELEMENTS   OF  ALGEBRA. 


For  his  six  dollars  in  money  will  cancel  only  six  dollars  of  his  debt, 
and  leave  him  still  owing  four  dollars;  we  may  consider  him  as  be- 
ing worth  four  dollars  less  than  nothing.  The  total  number  of  dol- 
lars  that  he  is  worth  may  be  represented  by  —  4,  because  it  will  take 
four  dollars  in  addition  to  the  six  dollars  to  pay  the  debt.  If  a  person 
gains  eight  dollars  and  loses  eleven  dollars,  the  number  of  dollars  in 
his  net  loss  may  be  represented  by  —  3,  because  it  will  take  three  dol- 
lars in  addition  to  the  gain  to  balance  the  loss.  Similarly,  if  he  gains 
100  dollars  and  loses  120  dollars,  the  number  of  dollars  in  his  net  loss 
may  be  represented  by  —  20.  To  enable  us  to  represent  these  num- 
bers, it  is  necessary  to  assume  a  new  series  of  numbers,  beginning  at 
zero  and  descending  in  value  from  zero  by  the  repetitions  of  the  unit, 
precisely  as  the  natural  series  ascends  from  zero.  To  each  of  these 
numbers  the  sign  —  is  prefixed.  The  negative  series  of  numbers  is 
written  thus  : 


10,  -9,  -8,  -7, 


6, 


-1,  0. 


For  convenience  the  algebraic  series  of  numbers  is  represented  as 
follows : 

Scale  of  Numbers.  We  may  conceive  algebraic  numbers 
as  measuring  distances  from  a  fixed  point  on  a  straight  line, 
extending  indefinitely  in  both  directions,  the  distances  to 
the  right  being  positive,  and  the  distances  to  the  left  nega- 
tive. From  any  point  on  the  line,  measuring  tovjard  the 
right  is  positive  and  tovmrd  the  left  negative. 


-^-h 


-lis 


-ilo 


-l5 


llO 


+  115 


In  the  above  illustration  consider  A  the  zero  or  starting-point  on 
the  scale  of  numbers,  and  the  distance  between  any  two  consecutive 
numbers  one  unit.     The  distances  to  the  right  and  left  of  A  are  posi- 


^ 


i 


FIRST    PHINCIPLES.  13 

tive  (+)  and  negative  (— ),  respectively,  as  indicated  by  the  direc- 
tions of  the  arrows. 

To  add  +  9  to  +  4  (read  9  and  4  in  tlie  positive  series) ^  we  start  at 
4  in  the  positive  series,  count  nine  units  in  the  positive  direction,  and 
arrive  at  13  in  the  positive  series.     That  is,  +  4  +  (+  9)  =  13. 

To  add  +  9  to  —  4  (i-ead  9  in  the  positive  series  and  4  in  the  negative 
series) f  we  begin  at  4  in  the  Jiegative  series^  count  nine  units  in  the 
positive  directioiiy  and  arrive  at  5  in  the  positive  series.  That  is, 
-4-f  (+9)  =  -+-5. 

To  add  —  9  to  -I-  4,  we  start  at  4  in  the  positive  series,  count  nine 
units  in  the  negative  direction,  and  arrive  at  5  in  the  negative  series. 
That  is,  +  4  +  (-  9)  =  -  5^ 

To  add  —  9  to  —  4,  we  start  at  4  in  the  negative  series,  count  nvie 
units  toward  the  left,  and  arrive  at  13  in  the  negative  series.  That 
is,  -  4  +  (-  9)  =  -  13. 

To  subtract  +  9  from  +  4,  we  start  at  4  in  the  positive  series,  count 
nine  units  in  the  negative  direction,  and  arrive  at  5  in  the  negative 
series.     That  is,  +  4  -  (+  9)  =  -  5. 

To  subtract  +  9  from  —  4,  we  begin  at  4  in  the  negative  series, 
count  nine  units  in  the  negative  direction,  and  arrive  at  13  in  the 
negative  series.     That  is  —  4  —  (4-  9)  =  —  13. 

To  subtract  —  9  from  +  4,  we  begin  at  4  in  the  positive  series, 
count  nine  units  in  the  positive  direction,  and  arrive  at  13  in  the  posi- 
tive series.     That  is,  +  4  -  (-  9)  =  +  13. 

To  subtract  —9  from  —4,  we  start  at  4  in  the  negative  series,  count 
nine  units  in  the  positive  direction,  and  arrive  at  5  in  the  positive 
series.    That  is,  -  4  -  (-  9)  =  -|-  5. 


13.  The  sign  +  is  often  omitted  before  a  number  in  the  positive 
series.  Thus,  the  numbers  3,  5,  and  6,  taken  alone,  mean  the  same 
^  (+3),  (-f-  5),  and  (+  6),  showing  that  the  numbers  are  in  the 
positive  series 

The  sign  —  must  always  be  written  when  a  number  is  in  the 
negative  series.  Thu.s,  the  numbers  3,  5,  and  6,  taken  in  the  negative 
series,  are  written  (-  3),  (  -  5),  and  (-  6). 

The  Algebraic  Signs  -f-  and  —  mark  the  direction  that 
the  numbers  following  them  are  to  take.     These  signs  are 


14  ELEMENTS    OF   ALGEBRA. 

used  to  indicate  opposition  (opposite  direction),  also  opera- 
tion. The  former  is  called  the  positive,  and  the  latter  the 
negative  sign. 

An  Algebraic  Number  is  one  -Which  is  represented  by  an 
algebraic  term  vnth  its  sign  of  direction.     Thus,  +  3,-3, 

—  a,  and  +  5  a  are  algebraic  numbers. 

Absolute  Value  shows  what  place  a  number  has  in  the 
positive  or  negative  series.  Thus,  +  3  and  —  3  have  the 
same  absolute  value ;  that  is,  three  units. 

Absolute  Numbers  are  those  not  affected  by  the  signs  -f 
or  —. 

Example.     The  meaning  of  an  algebraic  expression,  as 
3a;2+(-2a6)-[c-(-2/)], 
is  explained  thus  : 

To  3  x^  units  in  the  positive  series  add  2ab  units  in  the  negative 
series,  and  from  their  sum  subtract  the  expression  in  brackets,  c  in  the 
positive  series  minus  y  in  the  negative  series.  The  signs  written 
before  the  terms  (—2  a  6),  (—  y),  and  before  the  bracket,  indicate 
operation.  The  sign  written  before  2  ah  and  y,  also  the  sign  under- 
stood before  3  x^  and  c,  indicate  opposition. 

Exercise  6. 

1.  Over  how  many  units  and  in  what  series  of  numbers 
would  a  point  move  in  passing  from  4-3  to  —  8?  — 10  to 
+  1?  +5  to +15?  -12to-l?  -lto-12?  15  to  5? 
9  to  9  ?    —  5  to  —  5  ? 

2.  Which  is  the  greater,  0  or  —  6  ?    3  or  —  3  ?   —  5  or 

—  3  ?  +  10  or  —  1  ?  +  50  or  —  50,  and  how  many  units  ? 

3.  How  many  units  is  +  6  greater  than  0,  +  3,  —  3,  —  6, 
and  —  5  ?  How  many  units  is  —  5  less  than  5  ?  How 
many  units  is  a  less  than  b  ? 


FIRST   PRINCIPLES.  15 

4  If  a  point  start  at  (-f  3)  and  move  three  units  to  the 
right,  then  five  units  to  the  left,  where  is  it?  Express  its 
distance  from  0. 

5.  Suppose  a  point  start  at  (+  2),  and  move  six  units 
to  the  right,  then  eleven  units  to  the  left,  where  is  it  ? 

6.  Where  is  the  point  which,  starting  at  (—  5),  moved 
(—  3),  then  (+  8)  ?  Express  its  distance  from  the  starting- 
point. 

7.  Suppose  a  point  starting  at  +  3,  move  +  2,  then  —  7, 
then  -f  5,  then  —  6,  then  +  10,  then  —  1-1,  where  would  it 
be  ?    Express  its  distance  from  -I-  3. 

Explain  the  meaning  of: 

8.  2  [3  6  -f  (-  5  a)]  -  5  [(-  a)  +  (+  h)]. 

9.  (+  8a;)  4-  [+  3x-(+  12  y)  +  (-  x)  -  (8  y)]. 

Also  the  meaning  of  the  signs  +  (as  used  or  understood) 
and  — . 

Explain  the  meaning  of: 

10.  6«5A2  -f  (-\-a^}fi)  -f  (^aH^c^)  +  (-aH'^)  +  (+aH^) 
+  20  a2^c2  +  (-  aH^)  +  (-  a^b^). 

11.  aU^    ^     ^_  ^^.10)    ^     (^   ^6  ^5^     ^_     (_  y  a.)    +    ,;i3  ,^ft 

-  (4-  a^°  m^). 

12.  +  (+  a^)  ~{-\-h^)-  (+  a^)  -  (+  ^). 

13.  (x  +  7/)2  +  (a  +  xf  -  (x  +  7/)2  -  (a  +  x)^ 

Find  their  numerical  values  when  a^I?^c^m  =  n  =  k 
=  y  =  x=l. 


16  ELEMENTS  OF  ALGEBRA. 

Eead  the  following  expressions,  and  find  their  numeri- 
cal values  when  a  =  0,  b  =  1,  c  =  2,  d  =  S,  e  =  4, 
n  =  6,  and  m  =  6 : 

14.  c5  -  (+c)  -  (+  n^)  (+  c)  -  (+  d^)  +  a^(+  h^)  (+  <^\ 

15.  3  [g  +  (+  7ij\  -  5  (+  c)  -h  (a  +  6)  +  2  (+  «)  ^  6. 

16.  (+c)  [a  +  (4- ?i)  +  (+^)  -  (+6)]  -  (+  m)  -r  (+  d), 

17.  [(+m)  -  (+rO  +  (+m)  -  (+e)(+c)]  -  [(+m)  (+6)]. 

18.  d^^{^d')^'2{+h')-^{+c')-{+c^)-^d-e^-^{-\-4:% 

If  tt  =  5,  &  =  4,  c  =  3,  t^  =  2,  and  e  =  1,  find  the  numeri- 
cal values  of  the  following  expressions  : 

19.  (+a2)4-(+/^3)_(+c2)--(+e5);  a'^c-ahl?+d^-^{a'^W'). 

If  a  =  1,  5  ==  2,  c  =  3,  c?  =  4,  c  =  5,  find  the  values  of: 

26+2     3c-9     e2-l    d^    8a24.3?>2    4,2^552    c^^.^ 
e-3     6-2"^e  +  3'  F'    «2+Zy2  +  c2-62         ^2 


d>-h''       eh  c     '  b'^+d^-bd'  e^+ed+d^' 

a^  +  4a%+6a%'^  +  4:ab^-{-h\         28  12 


a^  +  Sa^b  +  Sal)^  +  b^     '  o?  +  h^j\-c^^ d?-c^-}p^' 

24.  aJ— 156-^5;  — ^ ^ ; . 

a  -\-  b  c  +  ft  o  +  c 

a*-4a3(?4-6«2c2_4r?.^3_|_^4    ,^       ,        ,^        ^,     ^^ 

25.  ,4    ^xq    .  ...22     .z.q 7  4;  12e-4a-^(2aX5)-26. 
b*—4:Wc+bb^c^—4:b(f-\-c^  ^  ^ 

26.  [(12e-4a)-^2«]  XZ^;  a^  ^  {aHU^)  ^  (^c^  -  a'^y 


29. 


FIRST   PRINCIPLES.  17 

27.  -T-i ;,  ,    >     +  ^   .    .  ,   , ;  («  +  ^)  (6  +  c). 

28.  (6  +  c)  (c  +  rf)  +  (c  +  rf)  (rf  +  e)  +  -^_^ji^_^. 

30.  66-^-(a  —  c)  —  3rf  +  a6ctZ-i-24a;c  +  5f]^-Ho 
+  a  X  e. 

Express  the  following  statements  in  algebraic  symbols  : 

31.  To  the  double  of  a  add  6. 

32.  To  five  times  x  add  h  diminished  by  one. 

33.  Increase  h  by  the  sum  of  a  and  x  divided  by  y. 

34.  Write  x,  a  times. 

What  is  the  sum  oi  x-\-  x-^x  ■\- ....  written  a  times ? 

35.  Write  three  consecutive  numbers  of  which  n  is  the 
least. 

36.  Write  five  consecutive  numbers  of  which  m  is  the 
greatest 

37.  Write  w,  a  minus  1  times ;  also  m  plus  n  times. 

38.  Write  seven  consecutive  numbers  of  which  x  is  the 
middle  one. 

39.  Write  a,  icth  power,  minus  y,  nth  power. 

40.  To  the  double  of  x,  increased  by  a  divided  by  &,  add 
the  product  of  a,  J,  and  c. 

41.  To  the  product  of  a  and  h  add  the  quotient  of  x  di- 
vided by  a,  and  divide  their  sum  by  y  diminished  by  c. 

2 


18  ELEMENTS   OF  ALGEBRA. 

42.  Write  a  exponent  n  plus  the  quotient  of  x  divided 
by  y,  minus  h  times  the  quotient  of  h  divided  by  the  ex- 
pression, a  exponent  c  plus  &  exponent  m,  is  greater  than  h 
minus  x, 

43.  Write  x  fifth  power  minus  h  sixth  power  ])lus  y  to 
the  ??ith  power,  divided  by  z  to  the  n\\\  power,  is  less  than 
g-  tenth  power. 

44.  Write  a  to  the  ??-th  power  divided  by  h  exponent  m, 
minus  x  exponent  n,  equals  a  minus  h,  divided  by  the  sum 
of  a  second  power  and  h  third  power. 

45.  Write  c  fourth  power  divided  by  a  second  power, 
minus  the  product  of  ^  and  y,  plus  x  ....  written  n  times, 
equals  a  exponent  m. 

46.  X  exponent  m,  plus  the  fraction,  a  fifth  power  minus 
three  times  a  second  power  h  third  power,  divided  by  x 
minus  y,  equals  x  minus  y,  added  to  the  sum  of  4  a  and  h 
minus  m,  plus  1  divided  by  x  to  the  nth  power. 

47.  Five  times  the  third  power  of  a,  diminished  by 
three  times  the  third  power  of  a  times  the  third  power 
of  &,  and  increased  by  two  times  the  second  power  of  h. 

48.  Three  times  x  exponent  2,  minus  twice  the  product 
of  X  exponent  3  and  y,  plus  the  third  power  of  a. 

49.  Six  times  the  third  power  of  x  multiplied  by  the 
second  power  of  y,  minus  a  exponent  2  times  the  fourth 
power  of  h. 

50.  a  times  the  second  power  of  n,  divided  by  x  minus 
?/,  increased  by  six  a  times  the  expression  x  plus  y 
minus  z. 


ALGEBKAIU    ADDllluN. 


19 


CHAPTER    ir. 

ALGEBRAIC  ADDITION. 

14.  In  Art.  12  it  was  shown  that  to  add  a  positive 
miniber  means  to  count  so  many  units  in  the  positive  di- 
rection, and  to  add  a  negative  number  means  to  count  so 
many  units  in  the  negative  direction. 

In  Algebraic  Addition  of  several  numbers,  we  count  from 
the  phice  in  the  series  occupied  by  any  one  of  the  num- 
bers, as  many  units  as  are  equal  to  the  absolute  value  of 
the  numbers  to  be  added  and  in  the  direction  indicated 
by  their  signs.     Thus, 

ExAMPLK  I.     Fiml  the  sum  of  3  a  and  —  9a 

Solution.  3  a  signifies  a  taken  3  times  in  the  positive  series,  and 
—  9a  aigniiies  a  taken  9  times  in  the  negative  series  We  count  from 
+  3  a,  9  a  units  in  the  negative  direction,  and  a  is  tiikeii  ia  all  6  times 
in  the  negative  serie.*?,  or  —6a.  That  is,  3a+  (— 9a)  =  -  6a. 
Similarly  (-}-  9  a)  +  (-  3  a)  =  4-  6  a. 

Example  2.    Find  the  sum  of  a,  2  6,  and  (—  3  c). 


>  + 


"N^ 

>i 

+Z^ 

A         D 

c 

o 
-3C 

< 

20  ELEMENTS   OF   ALGEBRA. 

Explanation.  Suppose  these  algebraic  numbers  to  be  accurately 
measured  as  represented  on  the  line  of  numbers  A  C.  Start  at  By 
then  count  2  b  units  in  the  positive  direction  and  arrive  at  G.  Now 
count  3  c  units  in  the  negative  direction,  and  arrive  at  D  in  the  posi- 
tive series. 

Thus,  (+  a)  +  (+  2  6),  or  a  +  2  6  =  ^  C  ; 

a  -{-  2b  +  (-  3  c),  or  a  +  2h  -  3  c  =  A  D. 

The  sum  of  the  algebraic  numbers  is  equal  in  absolute  value  to 
A  D  in  the  positive  series.     That  is, 

(+  «)  +  (+  2  6)  +  (-  3  c)  =  a  +  2  6  -  3  c.     Hence, 

The  sum  of  several  algebraic  numbers  is  expressed  by  con- 
necting them  loith  their  proper  signs. 

Notes  1.  The  sum  of  several  algebraic  numbers  is  the  excess  of  the  num- 
bers in  the  positive  series  over  those  in  the  negative  series,  or  the  excess  of  the 
numbers  in  the  negative  series  over  those  in  the  positive  series,  according  as 
the  one  or  the  other  has  the  greater  absolute  value.  Thus,  in  Example  1  the 
algebraic  sums  are  —Qa  and  +  %a.  In  Example  2  the  algebraic  sum  is  A  D 
in  the  positive  series. 

2.  The  sum  of  algebraic  numbers  is  the  simplest  expression  of  their  aggre- 
gate values. 

3.  Algebraic  addition  is  not  always  augmentation  as  in  arithmetic.  Thus, 
(+  7)  +  (-  5)  =  2  ;  also  (+  8)  +  ( -ll)  =  -  4. 

15.  A  Coefficient  of  a  term  is  d^  factor  showing  how 
many  times  the  remainder  of  the  term  is  taken.     Thus, 

In  the  term  5  abm,  5  is  the  coefficient  of  a b m,  and  shows  that 
a 6 m^s  taken  5  times  ;  5  a  is  the  coefficient  of  6 m  ;  5  ab  is  the  co- 
efficient of  m.  In  the  term  4  m  (a  6  -  2  a),  4  is  the  coefficient  of 
m  (rt  6  —  2  a)  ;  4  w  is  the  coefficient  of  (a  6  —  2  a). 

Note.    A  coefficient  may  be  numerical  or  literal.    When  no  nnmerical 

coefficient  is  expressed,  1  is  always  understood  to  be  the  coefficient;  as,  x  ;  xy°. 

Like  Terms  are  those  having  the  same  letters  affected 
with  the  same  exponents.     Thus, 


ALGEBRAIC  ADDITION. 


21 


2m^ujc^  </*,  m"^  n  jfi  y*,  and  —  10  m*  n  x*  y^  are  like  terms,  as  are 
also  bx^^y^z*  and  —  3x"y*z*;  but  dx*y^  and  5x*i/*2*  are  un- 
like terms.     Like  terms  are  said  to  be  similar. 

A  Monomial  or  Simple  Expression  consists  of  one  term;  as, 
x:  lU  ((  l>c:  —  5  a^  j^. 


16.   To  Add  Similar  Monomials. 

I.  When  all  the  Terms  are  Positive  or  Negative.  Add 
the  numerical  coefficients;  to  the  sum,  annex  the  common 
symbols^  and  prefix  the  common  sign. 

II.  When  Some  of  the  Terms  are  Positive  and  Some  are 
Negative.  Add  separately  th£  numerical  coefficients  of  all 
tlie  positive  terms  and  the  numerical  coefficients  of  all  the 
negative  terms;  to  the  difference  of  these  two  results^  annex 
the  common  symbols,  and  prefix  the  sign  of  tlie  greater  sum. 


EXAMF'LE  1.     Find  the  sum  of  10 a; y*,  -  Sxy^  4x?/*, 
and  —\lxy^. 

Explanation.  For  convenience  write  the  terms 
as  shown  in  tlie  margin.  The  sum  of  the  coeffi- 
cients of  the  positive  terms  is  14,  and  the  sum  of 
the  coefficients  of  the  negative  terms  is 'SI.  The 
difference  of  these  is  17,  and  the  sign  of  the 
greater  sum  is  negative.  Hence,  the  required 
sum  is  —  nxy*. 


Uxf, 


Process. 

+    Axy^ 

-  3xy^ 

-  Uxy^ 

—  M  xy^ 

—  17x2/* 


Example  2.    Find  the  sum  of  (x  -f  y),  1.1  (ic  +  y),  -  2.9  (x  +  y), 


.29  (x  +  J/) .  -  i  (x  +  y),  and  1 .26  (x-\-y). 

Explanation,  (x+y),  enclosed  in  parentheses, 
is  treatt^l  as  a  simple  symbol.  The  coefficients 
of  (x  + 1/)  are  1,  l.l,  2.9,  29,  i,  and  1.26.  The 
sum  of  the  coefficients  of  the  positive  terms  is  3.65, 
and  the  sum  of  the  coefficients  of  the  negative 
terms  is  3.15.  The  difTerence  of  these  is  .5,  and 
the  sign  of  the  greater  sum  is  positive.     Etc. 


Process 

+  C-^  +  y) 
1.1  (x  +  y) 
+  .29(x-f-2/) 
-f  1.26  (x  4- 2/) 
-2.9    (x  +  y) 

-    H^  +  y) 

+  .5(x-fy) 


22  ELEMENTS  OF  ALGEBRA. 

Exercise  7. 

Find  the  sum  of: 

1.  (+  2  a),  (+  a),  (+  4a),  (+  3a),  (+5 a),  and  (+  la). 

2.  (+  5  a  a?),  (+  2  a  x),  (+  6  a  x),  (+  a  x'),  and  (+  a  x). 

3.  (+  6  c),  (+  8  c),  (+  2  c),  (+  15  c),  (+  9  c),  and  (+  c). 

4.  (-6a&c),  (+4«&c),  i+ahc),  (-2abc),  and  (+5a&c). 

5.  (-fa;2),  (-|^^2)^  (-1^2)^  (-i^''),  and  (-x^). 

6.  (+  f  ^),  (-  I  ^),  (+  I  ^\  (-  2  ^),  (+  I  x),  and  (+  ^). 

7.  (+  3  a3),  (-  7  a3),  (-  8  a«),  (+  2  a^)^  and  (-11  a^). 

8.  (+  4a2?>2)^  (_  aH2),  (-  7  a^h^),  and  (+  .5a2&2). 

9.  (+7ahcd),  (+  2ahcd),  (-l.labcd),  and  (-4.1aZ>ctQ. 

10.  +  (ft  -^  6'),  -  .01  {b  +  c),  +  .7{b  +  c),  -10  (b  +  c), 
and  +  i(b  +  c), 

11.  +10(,.— 2/)3,  -(x'-2/)3,  +m(x-yf,  -2{x-y)\ 
and  —  3  (ic  —  ?/)^. 

17.  If  the  monomials  are  not  all  like,  combine  the  like 
terms,  and  write  the  others,  each  preceded  by  its  proper 
sign  (Art.  14). 

Example  1.  Find  the  sum  of  (+  7  k),  (^-  3  h  if),  (-  2  x),  (-5  6  y% 
(+  4  X),  (-8  6  /),  (+  9  x),  (+  6 1/),  (+  1 1  x),  and  (-  h  y^). 


ALGEBRAIC   ADDITION. 


23 


Ilzplanation.  For  convenience,  write 
the  expressions  so  that  like  terras  shall  stand 
in  the  same  column,  as  in  the  mai-gin. 
The  sum  of  the  terms  containing  x  is 
-f  29  X,  and  the  sum  of  those  containing 
b  y^  is  —  10  6  y"^.  Hence,  the  result  is 
-H  29x-  10  6  2/'*. 


Process. 

4-  7x4-  3  6  1/2 
-  2  X  -  bhy^ 
+  4  X  -  8  6  1/2 
+  9x-f  6 1/2 
+  llx-  6y2 
+  29  X  -  10  6  1/2 


Example  2.     Find   the  sum  of   -f  .05  (a  -|-  6),   -  .01  (m  +  n) 
4-  7  (a  +  6),  -  3  (m  +  n),  -  1 1  (a  +  6),  and  -f  10  (m  -h  n). 

Explanation,  (a  -f  &)  and  (m  +  w), 
enclosed  in  parentheses,  are  treated  as 
simple  symbols.  The  sum  of  the  like 
terms  containing  (a+/>)  is  —  3.95(a-|-ft). 
The  sum  of  tiie  like  terms  containing 
(m +  «)  is  +  6.99  (m  +  n). 


Process. 

-f    .05(0  +  6)-    .01(m  +  «) 

+       7  (a +  6)-      3(m4-w) 

-      11  ((!  +  ?>)+    10(m  +  yi) 

—  3.95  (a  +  6)  -f  6.99  (m  +  n) 


Exercise  8. 
Find  the  sum  of : 

1.  (+  .3  X),  (+  .5  y\  (+  .01  .>),  (+  3  y\  and   (-  7  x). 

2.  (4-  I  a),  (-  J  a  J),   (+  §  a),    (-h  ^  rr  ft),  and   (-  |  ..)• 

3.  (+  5  (-2 ./2),   (-  2  a«./;),  (-  2  c2.r2),  (+10  a^  x),  (+  8  c2  x^), 
(-4a«x),  (-4c2.x'2),  and  (-f  4a8a?). 

4.  (+  I  .^),  (-  J  a  5),  (+  V  •^').  (+  1^5  ^^  ^').  (+  A^^)> 
(+^^').   (-HJ«^).  (-i^^),  {-i^^'').  and  (+§«6). 

5.  7a,  -  :^x-yl  8a,  .3 (x-?/),  .03Cr  -  y),  and -.la. 


18.   A  Polynomial  <>i  Compound  Expression  consists  of  two 
or  more  terms. 


24  ELEMENTS  OF  ALGEBRA. 

Example.  Find  the  sum  oi  8ax—  .ly  +  5,  .7ax  +  y  —  am  —  9f 
and  -  .3ax-  1.02  y  +  5p-  .3. 

Process.     Sax  —      .1  y  +5 

.7  ax  +  y  —  am  —  9 

-.Zax-  l.02y  ~    .3-t-5p 

8Aax  —    .12  y  —  am  —  4.3  +  b  p      Hence,  in  general, 

To  Add  Polynomials.  Write  the  expressions  so  tliat  like 
terms  shall  stand  in  the  same  column.  Find  the  smn  of  the 
terms  in  each  column,  and  connect  the  results  with  their 
proper  signs. 

A  polynomial  may  be  regarded  as  the  sum  of  its  monomial  terms. 
Thus,  the  sum  of  the  terms  (-f  <?),  (—  h),  and  (—3  c)  is  «  —  ft  —  3  c. 
Hence,  the  sum  of  two  or  more  polynomials  whose  terms  are  all 
unlike  is  expressed  b}'^  writing  their  terms  ivith  their  respective  signs. 
Thus,  the  sum  of  a  —  b,  c  —  d,  and  m.  +  n  —  x  is  a  —  b  +  c  —  d 
+  m  +  n  —  x. 

Exercise  9. 

Find  the  sum  of :       • 

1.  2x-{-y,  bx+3y,  — 3x  —  2y,  and  —  4 a^  +  Sy. 

2.  5x  +  Sy+3a,  —7x  +  4:y  —  8  a,  and   2  x  —  3  y. 

3.  3  6-3^,  2  c  -  2  ^,  3c -71),  and  4:h-2c  +  3x. 

4.  14a  +  x,  I3h-y,  -11a  +  2  y,  and  ic  -  2  a  —  12  &. 

5.  ax  —  4:mn  +  hd,  hd  —  a  x  —  3  mn,  7  m  n  —  3  ax 
+  3hd,  and  5mn  —  30  ax  —  9hd. 

6.  a  —  h,  21)  — c,  2  c  — d,  2d  —  3(f-\-n,  and  m  —  n  +  x. 

7.  a  b  c  -\-  3  a  b  m  —  5  c  m,  3cm+  11  abm-\-9abc, 
90abm  — 21cm  — 31  abc,  and  3cm  —  51  abm-\- 13 abc. 

8.  T/i  +  n  +  p,  m  —  n—p,  m  —  n+  p,  and  m  -\-7i—p. 


ALGEBRAIC  ADDITION.  25 

and  —  a  +  6  +  f  4-  rf. 

10.    1.25  a 6  +  1.1  c  +  99  h,  ami    3  ah  ■{■  2.2  c  -\-  1.01  i. 

11  ia^iah  +  .QV',  la  -  ^^^  «  ^  "  1^  ^'.  I  ^'  +  i^^2> 
f  .6  ^,  and  .1  a  -  1.01  «  h 

12.  J  ?/|2  -  .2  wi  -f  J,  .1  wi2  +  .01  m  -  2,  m^  _|.  3  j  ^^^  ^^  _  ^^ 
and  J  m  —  5S  m  71  —  1 J  w^  —  2|. 

13.  xy  —  ac,  Sxi/  —  9a(\  —7xy  +  5ac+lMcd, 
4  xy  -^  6  fl  c  —  .09  f  rf,  and  —  x y  —  2  a  c  +  c  d. 

14.  .5  r(3  -  2  r|2  h  -  I  b\    lu^b-  .75  a  l^  +  2  b^   and 

15.  3  (m  -  nf  +  .3  (a:  +  ?/)^  .4  (m  -  7/)3  -  .2  (x  +  y)\ 
.7(m-n)8-3.03(a;  +  i/)^  and  5.1  (m- 7^)3 - M 1  (.aH-.#. ' 

16.  oa^V^-^a^b^-\-x^y-^xf,  4a^b^-7 a^ b^-'Axf 
+  (jx^y,  3a^l^-^Sa^b^-3a^y+5x  y\  and  2  a^lfi 
^a^JJi  ^  3  a^  y  —  'S  X  if. 

17.  |aa.-2-§/^2  4.  |.,3y^.  j8^  3^.x^+ |.r^?/+ 7.5^2 
+  J  fcs    2  ^  .^2  ^  3  ,3  ,j  _  1  ,,2  _  1 2,8^  j^.^a   Jj.  a  x^-V\x^  y 

18.  ac^  ^  \a}?'  -V  In"^  -  \a^b  -  \abc\  \a^  c,  \nn 
j^lb^^laV^^  \b<?-\-2ahr  +  \\\?c,  and  1.1  ^rc-  1  2ac2 
+  i^^c-  |/>6-2l  l..Sc3+  1.23  r/ 6c. 

19.  3.1«8-4.2a?2+1.2a;  +  1.7,  2.22  a,-3- 1.2 a^  +  3.33 x- 

-  10.09,    2  a^  +  7  a^  -  2  .r  +  1,    3  ./;8  +  1.22  a^»  +  12.12 

-  1.33  a;,  and  11.1 1  j?  +  5.55  x^  -  0.2  u.-  +  3.77. 

20.  «H2c3  +  «2^^,2  ^  3  ^^2/^3  ,5     13  ,,3^2,;3  _  1  4  ,,2^,3^8 

+  1.5  a2  63  c2,    1.5  «2  63  c2  -  1.9  ^3  }?•  c^  ^  1  3  ,,2  ^3  ,5    and 
1.7  a3 62(^3  _  1.2^(2^^5  4.  101  a^ftSca. 


26  ELEMENTS  OF  ALGEBRA. 

21.  2c'"+.la"  +  3&c,  .5c'"  +  3.9ft"+2.02&c,  c'"  +  2.09a'» 
-|&c,  and  i&  +  f  Z^c-  3.03. 

22.  a  &  —  a:'  +  |-  a??/,  .1  X  -{-  .01  a  ic  —  2.02  a  6,  ^  ax 
■\-  ^xy  —  ^ah,  6ic  ~  1.01  ax  +  .la;,  and  ■J'^'  2/  +  j  a  x 
-\xy, 

23.  3  ^  +  .ly-  7.01  ^  +  6.01  ?/  +  .2  a  +  3  ^  -  1.5  c; 
-.8^  +  9.01a  +  3.03?/  +  ^-4.04?/-  2.01 2;+  2.2  a -y. 

24.  3  a  6  4-  9  -  ic2  ?y,    a;3  3/  +  3  a;  2/  +  5,  6  a;^/^  +  4  a;2y 

—  3  icy,  10  o^y  +  1  +  3  a;  2/2,  and  17  —  3  x^y  —  2  a;^y. 

25.  .5  {m  -  3xy,   -  |  (m  -  Sx)\    .75  (m  -  3  x^,  and 

—  1.25  (>yi  -  3  xy. 

26.  ^2  +  Z;4  4.  c3,  _  4  tt2  -  5  c^  8  a2  _  7  &4  +  10  c3,  and 
6  &4  -  6  c3. 

27.  3a2-4a&  +  &2+  2a  +  3&-7,  2^2-4^2  +  3a 
-56+8,    10  ab+  Sb^  + 9  b,    and    5  ^2  _  6  a  &  +  3  ^2 

+  7  a  -  7  &  +  11. 

28.  x^  —  4:a^y  +  6x^y'^  -  4:Xy^  +  /,  4:S(^y  —  12x^y^ 
+  12  xy^  —  Ay\  6  x^y^—12xy^  +  6  y^,  and  4:xf  —  4:y\ 

29.  a3+  «62+  ac^-a^b-abc-a^c,  a?b  ^- b^  +  bc^ 

—  ab"^  —  b'^c  —  ab  c,    and    a^  c  +  b"^  c  +  c^  —  a  b  c  —  b  c^ 

—  a  c2. 

30.  5  a^  -  2  a^b  +  9  ab^  +  17  b^,  -2  a^  +  5  a^  b 
-4:ab^-12b^  b^-4:ab^-5  a'^b-a^  and  2  ^2  6 
-2  a^-Ob^-ab"^. 

31.  a;»«  — ?/"+3<  2ic'"-3/-a,  and  a;'»  +  42/"— a''. 


ALGEBRAIC   SUBTRACTION. 


27 


CHAPTER  III. 
ALGEBRAIC  SUBTRACTION. 

19.  In  Art.  12  it  was  shown  that  to  subtract  a  positive 
imuiber  means  to  count  so  many  units  in  the  negative  di- 
rection, and  to  subtract  a  negative  number  means  to  count 
so  many  units  in  the  positive  direction.  Hence,  the  addi- 
tion of  a  positive  number  produces  the  same  result  as  the 
subtraction  of  a  negative  number  having  the  same  absolute 
value. 

Thus,  4-3+ (+6)  =  +  3  +  6  =  9.     +3- (-6)  =  +  3  +  6  =  9. 

Also,  the  subtraction  of  a  positive  number  produces  the 
same  result  as  the  addition  of  a  negative  number  having 
the  same  absolute  value. 

Thus,  +  4-(+6)  =  f4-6  =  -2.     +4+(-6)  =  +4-6  =  -2. 

We  observe  that  the  subtraction  of  one  number  from 
iinother  produces  the  same  result  as  counting  or  measuring 
from  the  place  occupied  by  the  subtrahend  to  the  place 
occupied  by  the  minuend.     Thus, 

Subtract  —  h  from  +  a  ;  also  +  a  from  —  6. 


-a^ 


-<r 


-b 

a-b 


+b 


a<r 


IB 


D 


28  ELEMENTS   OF   ALGEBRA. 

Explanation.  Suppose  the  algebraic  numbers  to  be  accurately 
measured  on  the  line  of  numbers  C  D.  We  start  at  B  in  the  positive 
series,  count  b  units  in  the  positive  direction,  and  arrive  at  D  ;  and 
the  distance  from  A  (0)  to  Z>  is  equal  in  absolute  value  to  A  D  in  the 
positive  direction.  But  in  counting  from  ^  to  Z>  the  absolute  value  is 
the  same  as  the  absolute  value  in  counting  from  C  (the  subtrahend) 
to  B  (the  minuend),  and  we  have  counted  in  the  direction  opposite  to 
that  indicated  by  the  sign  of  the  subtrahend.     Thus, 

C  B  =  -h  {+b)-\-  (+a)  =  a  +  b.     That  is, 
(+  a)  -(-b)  =  a  +  b. 

Subtracting  +  a  from  —  b  gives  the  same  result  as  counting  from  a 
in  the  positive  series  to  b  in  the  negative  series,  and  the  distance  from 
5  to  C  is  equal  in  absolute  value  to  B  C  in  the  negative  direction. 
Thus, 

BC  =  +  (-a)  +  (~b)  =  -a-b.     That  is, 

(—  b)  —  (-{-  a)  =  —b  —  a,  OT  —  a  —  b.     Hence, 
Algebraic  Subtraction  is  the  operation  of  finding  the  dif- 
ference from  the  subtrahend  to  the  minuend. 

To  subtract  —  5  a  from  +  2  a  is  the  operation  of  finding  hoio  far 
and  in  ivhat  direction  we  must  go  to  pass  from  5  a  in  the  negative 
series  to  2  a  in  the  positive  series,  and  is  found,  by  counting  from 
—  5  a  to  +  2  a,  to  be  7  a  units  in  the  positive  direction.     That  is, 
+  2a  -  (-  5a)  =  +  7a. 

To  subtract  +  5  a  from  -  2  a,  we  count  from  5  a  in  the  positive 
series  to  2  a  in  the  negative  series  and  pass  over  7  a  units  in  the 
negative  direction.     That  is, 

-  2  a  -  (4-  5  a)  =  -  7  a. 

These  differences  may  be  found  by  changing  the  signs  of  the  sub-' 
trahend  and  proceeding  as  in  addition,  as  shown  by  a  comparison  of 
results.     Thus, 

Minuend.  Subtrahend.  By  Addition. 

+  2a-(-5a)  =  +  7a      >|  j^+2rt  +  (-t-5«)=4-7a. 

-2a- (+5a)=-7a        '  I   -  2  a -f  (- 5  a)  =  -  7  a. 

+    a  -  (-      &)  =  +  a  +  6    f  '''''  )    +     a  +  (-{-     b)  =  a  +  b. 

-    b-(+     a)  =-a-b  )  \^-     6+(-a)=-a-6. 


ALGEBRAIC  SUBTRACTION.  29 

Hence,  in  general, 

To  Subtract  one  Algebraic  Number  from  another.  Change 
the  sign  of  the  subtrahend,  and  add  the  result  to  the  minuend. 

Notes:  1.  Algebraic  subtraction  considered  as  an  operation  is  not  distinct 
from  addition  ;  for  it  is  equivalent  to  the  algebraic  addition  of  a  number  with 
the  opposite  algebraic  sign.  It  includes  not  only  distance  but  direction,  and 
direction  depends  upon  the  sign  of  the  subtrahend  and  which  number  is  consid- 
ered the  minuend. 

2.   Algebraic  subtraction  is  not  in  all  cases  diminution.    Thus, 
8  ~  (-  2)  -  10 ;  also  2  -  (-  8)  -  10. 

E.xAMPLE  1 .     Subtract  +  3  o* 6  c  w*  from  -f  10  a*  6  c  7/1^. 

Solution.  Changing  the  sign  of  the  subtrahend,  and  proceeding 
as  in  adtlition,  we  have  +  10  a*  6  c  m^  -f  (—  3  a*  6  c  m*)  =  +  7  a^bcm^. 

Example  2.     Subtract  +  27  (x^  -  i/«)»  from  13  (x^  -  i/)». 

Solution.  Treating  (t*  —  y^y  as  a  simple  symbol,  changing  the 
sign  of  the  subtrahend  and  proceeding  as  in  addition,  we  have 

13  (x2  -  7/)«  +  [-  27  (x^  -  !/«)»]  =  -  14  (z"  -  y«)». 

Exercise  10. 
From : 

1.  +OaHc  take  —aHc,  —Ual^xy^  take  +19 al^xi/. 

2.  +:tVtake-./.y;  +99  mnp^rst^^  take  ■}-99mnphst^^. 

3.  —10  axy  take  —axy\  x'^  take  —he. 
From  the  sura  of: 

4.  —  11  a.-^,  +  .5  d^,  and  +  1.25  J"  take  -f  5.5  0^. 

5.  ah^,  —  Sabc^,  and  +  .3  ah(^  take  the  sum  of 
-abc?^,  +  SMabc^  and  -  IMabc^. 

6.  lOS  mnp^^,  —lO.Smnp^^,  and  +vinp^^  take  the 
sum  of  -- 10  771  np^^,  +  33  mnp^^,  and  —  108.1  m  np^^ 


30  ELEMENTS   OF  ALGEBRA. 

7.  5  {x  +  y),  —  2{x-\-  y),  and  +  {x-\-y)y  take  the  sum  of 

-  {x  +  v/),  +  6  {x  +  2/),  and  -  2.5  {x  +  ?/). 

Find  the  aggregate  value  of: 

8.  +  17  a  x^  -(-5a  a:-3)  +  (-  24  a  ^'S)  -  (+  a  j:^). 

9.  +  19  a  ^  ?/2  +  (+  a  ;2;  7/2)  —  (—  5  a  x  ?/). 

10.  +  x-2y  +  (-  x^y)  -  (+  x^y)  -  (-  ^2y)  +  (_  1  .^^2^/) 

-  (+  3  x^y)  -  (-  10a;2y)  +  {-bx^y).  ' 

11.  +  n«  +  ^)'  -  [-  -1  («  +  ^)']   +   [-(«  +   ^)'] 

-  [+  I  («■  +  ^)']  +  10  (a  +  ?^)2. 

12.  I  .-  +  (+  1  X-)  +  (-  .1  X-)  -  (+  ,2^-  ^")  +  (-  1  x^ 

+  (+ 1.  ix'')-{-2>ix'y 

20.  Example  L  Subtract  2  ah  +  f)  a^  if  -  U  a^  -  1  y^  from 
15  a3  -  8  1/  +  23  a3  t/S. 

Process. 

Minuend ]  5  a^  _  8  ?/3  +  23  a^  i/S 

Subtrahend,  with  signs  changed       +  14  a*  +  7  ?/^  —    5  a^  ?/^  —  3  a  6 
Difference 29  a^  -     ^/^  +  18  a^  ^/S  -  3  a  6 

Example  2.     Subtract  3  a;  ?/  +  n  -  5  a^  6  +  5  jo''  from  5  ic  i/^  -  3  a^  5 

+  3  771. 

Process. 

Minuend 5  xy^  -  3  a'^h -\- Sm 

Subtrahend,  with  signs  changed     -3a:  7/^+5  a- 6  —  n  —  b  p^ 

Difference 2xy^-\-2a^h+'3m  —  n  —  5p^ 

Hence,  iu  general, 

To  Subtract  one  Polynomial  from  another.  Change  the 
algebraic  sign  of  every  term  in  the  subtrahend,  and  add  the 
result  to  the  minuend. 

Note.  It  is  not  necessary  that  the  signs  of  the  subtrahend  be  actually 
changed,  we  may  conceive  them  to  be  changed. 


ALGEBRAIC   SUBTRACTION.  31 

Exercise  11. 
Subtract : 

1.  5x~3y-f22  from  3  x  -\-  y  —  z. 

2.  X  —  y  •{■  z  from  —  x  —  2  ij  —  3  2. 

3.  a  —  6  4-  20  6'  from  —  a  —  h  •{•  10  c. 

4.  J^  —  iy  —  i^  from  \x  -\-  y  +  z. 

5.  lx-\y-\-^zhom-lx+t^y-^^z, 

6.  J^-iy-i  from  -Jx--Jy+  J. 

7.  a3  _  4  a2  J  ^  2  i  c  +  5  from  3  a^  -  «U  -  5  c  -  5. 

8.  a^y  —  Z  ah X  -\-  2  X y^  —  \  from  S  x^y  —  abx-^  2  xy^. 

9.  ahcxy  -\-2ahy  —  4ihx  from  2abcxy —  ab y  -\-  bx  —  'S. 

10.  acy  —  bxy  +  aJc  —  1  from  bxy  —  4:acy  —  abc  -fa. 

11.  .4a:*-.3a:3^.2ic2-7.1a;  +  9.9  from  a;^  -  2.10  a:^ 
+  .2ar»-:.07a.'  +  .9 

12.  1.2  /-s  -  1   4-  X  +   1.1  a;4  +  1.7  a^  +  a  from  \-x 
-.l.r*  +  .2.'/^-.3A'8  4-a. 

13.  \  vin^  ^In^-  yi^  +  I  w2  from   |  /7i3  -  J  r/i  7i2  -  ;,2 

14.  .125  m3  -  M^  ,n  vi2  _  .33^  ^  ^2  _  (5o|  ^3  _^  ^.  |Vom 

g  Wl^  —  I  Wl  n^  —  I  ?//,2  7J,  -I-  J  7^3 

15.  J  ,;i2  _  I  y  _  I  ^  4.  J  ^    |.^Qj^    ^  W2-  ^  y  4-  f  ?l  -  i  .^. 

16.  a^hc  +  xf  -Ic  from  3^  a2ftc  +  2Jaj?/"- 4f  c. 

17.  lOc  —  a-h  -^  b(l-\-  6rt-15c4-  3(Z  from  25«-5 
—  5c4-8r?  —  20rt. 

18.  af"  —  3  a:"  if  —  if  from  4  x*"  +  .c*  y"  _  af . 


32  ELEMENTS  OF  ALGEBRA. 

19.  —  .9  a"*  x^  -  .3  a  63  a?  +  .6  +  .03  IT  c  x^  from  .9  a'^x^ 
+  1.3  -2ah^x-\-  AlTcx^ 

20.  From  the  sum  of  lo?-\h^  +\  c\  J  cv^  -  3^  c\ 
and  2|  63  -  I  a2  _  i|  c^  take  ^i  a?  -  ^\  63-4  c\ 

21.  From  the  sum  of  3  ic3  _  ^2  ^  _  ^^^  7  ^3  _  il  ^2;3 
-f-  llj  y^  oc,  and  11  ;;c2;3  _  gi  ^3  _  2  1/2 a;  take  a;2;3  _  25  ^/^j; 
4-  1  ^3. 

22.  From    of  +  y""   take   the    sum   of   11  ^-^  +  y^  —  z, 

—  6  ^"  —  5  /*  —  3  ;2,  and  -  5  ^"  +  3  2/"*  +  4  z. 

23.  Add  the  sum  of  S^y-  .3  ^/^  and  5-3y  +  2.7  2/3 
to  the  difference  obtained  by  subtracting  3  +  l^y^  —  .5y 
from  1—2/3. 

Queries.  Why  change  the  signs  of  the  subtrahend  in  subtracting  ? 
Wliy  add  the  subtrahend,  with  signs  changed,  to  the  minuend  ? 
Does  the  use  of  the  signs  +  and  —  in  Algebra  differ  from  their  use  in 
Arithmetic  ?    How  ? 

Miscellaneous  Exercise  12, 

1.  From  m^  —  n  —  1  take  the  sum  of  2  n  —  3  +  2  ?^3 
and  3  ??i3  —  4:  +  5  'n?  —  n. 

2.  From  the  sum  of  1  -  8.8  y  +  .9  a^  and  1.1  x^  +  S  x^ 

—  .2  2/  —  1  subtract  2  x^  —  x^  +  5  y. 

3.  Take  x^  +  x  —  1  from  2  x^,  and  add  the  result  to 
-2a^-x^-x  +  l. 

4.  Take  a^  —  h^  from  ah  —  h^,  and  add  the  remainder 
to  the  sum  of  a6  -  ^2  -  3  6^  and  ^2  +  2  62 

5.  To  the  sum  of  m  +  7i  —  3  /?  +  5  and  2  m  +  3  ?i  +  5^ 

—  3  add  the  sum  of  m  —  in  —  7 p  and  5 ^  —  6  ??i  —  2. 


ALGKBRAIC    SUBTRACTION.  33 

6.  Take  3  a-^"*  -  2  x-2"  f^  -  y^-^  from  3  f/—'  +  2  x^'^y^ 

7.  Take  2x^y^-*6z^-\-2y^-V  z-1  from  a;iy*+2;--4yi 

8.  Take    .8  h^  y^  -  A  a?  x^  +  .3  rf   from    .4  a?  x^  +  .3  c 

-  1.2?)tyi 

9.  Take  f  xt-f  a;*y*H- 33j?/§  from  f  j:;^  +  |-^*?/*  + Jyi 

10.  From  the  sum  of  .7  c  y^  —  .4  a  x -f  .5  h,  .04  6  —  |  cy» 
+  i  w,-|a.r+ Jcy§-|,  and  -yi  a  x- .23  6- .8  m  +  .3 
take  the  sum  of  .55  a  ic  +  J  /?i  +  ^^  and  .33  in  —  1.1  cy^ 
+  .67  ?«. 

11.  Find  the  sum  of  «"•  -  7  6*  -h  cp  and  |  6"  +  |  a"*, 
and  subtract  the  result  from  cp  —  4:n. 

12.  From  a*  -  2  c"  -  af*  take  the  sum  of  J  «"•  -  ^  i" 

-  X'  and  i  a"*  +  J  6"  —  y"  —  af. 

13.  From  -  a*-6*-c'-^«  take  the  sum  of  Jft'"+  §6* 

-  I  C,  ^5  a"*  -  \l  c^,  and  V  ^^  +  ^'■ 

14.  From  3  (a^  -  ^y  -  (o^-a  +  ff  take  §  (^  +  2^)* 
-  c^  +  3J  («^  -  ^)*. 

15.  From  unity  take  3  a^  —  3  a  +  1,  and  add  5  a^  —  3  a 
to  the  result. 

16.  Add  3  ^-  -  7  a:"  +  1  and  3  a;8-  -f  ^  -  3,  and 
diminish  the  result  by  x^"*  —  2. 

17.  From  zero  subtract  I  a^  —  ^  x  -\-  2. 

18.  From  .3  m'  -  1  +  J  w  take  5  n^  -  2.7  m^  -  J  n, 
tlien  take  the  difference  from  zero,  and  add  this  last  result 
to  -  5  n^  4-  3.3^  m^  -f  n. 


34  ELEMENTS  OF  ALGEBRA. 

19.  What  expression  must  be  subtracted  from  10  y'^  -]-  y 

-  1  to  leave  3  y^  -  17  y  +  S  ^. 

20.  What  expression  must  be  subtracted  from  a  —  o  x 
+  y  to  leave  2  a  —  o  x  -\-  yl 

21.  From  what  expression  must  a^—bah  —  lhc  be 
subtracted  to  give  a  remainder  b  a?  -\-  3  ah  -\-  1  h  c1 

22  From  what  expression  must  a^  h^  —  b^  c^  +  6  a""  c** 
be  subtracted  to  leave  a  remainder  b^  c^  —  6  aJ^  c"  ? 

2*3.  To  what  expression  must  |  ft^  +  2\a  —  1^  a^  —  3 
be  added  so  as  to  make  2\  a^  —  2\  a  -\-  3\  a^  -\-  ^  ^ 

24.  To  what  expression  must  b  x y  —  lib c  —  1  mn  be 
added  to  produce  zero  ? 

25.  What  expression  must  be  added  to  3  ^"  —  3  ^""^  +  2 
to  produce  ^"  +  x^"~^  —  6  ? 

26.  What  expression  must  be  added  to  m  a?"*  —  ^"  +  2 
to  produce  m  a?'"  —  2  ? 

27.  From  the  sum  of  .6  {x  +  y)^  +  .3  a"  +  ^'",  |  a**  x"^ 

—  c^  ~  I  («  +  ?/)2,  and  I  (^  +  2/)^  ~  I  ^"  ^''"j  take  the  sum 
of  .3  (x  +  i/)^  -  I  a"  ;:c^,  I  «"  ic'"  -  6.5  (ic  +  y)^  +  c^  and 
ro  (^  +  .?/)^  +  3.3  a"  ^"^  —  3. 

Algebraic  Subtraction  may  be  defined  as  the  operation  of 
finding  a  number  which  added  to  a  given  number,  will 
produce  a  given  sum.  The  sum  is  now  called  the  min- 
uendy  the  given  number  is  the  subtrahend^  and  the  required 
number  is  the  difference. 


ALGEBRAIC   MULTIPLICATION.  35 

CHAPTEK   IV. 
ALGEBRAIC  MULTIPLICATION. 

21.  Evidently  dm  x  6n  =  5  x  (^  x  m  x  n  =  SOmn. 
Hence,  in  Algebra,  the  product  is  tlie  same  in  whatever 
order  the  factors  are  written. 

a  X  a  X  a  X  a  or  aaaa  is  written  a*,  and  shows  that 
a  is  taken  four  times  as  a  factor.  aXaXaXaxaov 
aaaaa  is  written  a^,  and  shows  that  a  is  taken  five  times 

as  a  factor,     a  X  a  X  a  X ton  factoi*s,  or  a  aa to  ?i 

factors  is  written  a",  and  shows  that  a  is  taken  n  times  as  a 
factor.     Hence, 

An  Integral  Exponent  shows  how  many  times  a  number 
or  term  is  taken  as  a  factor. 

a^  is  read  a  second  power,  or  a  exponent  two,  or  a  square. 
a'  is  read  a  third  power,  or  a  exponent  three,  or  a  cube. 
Hence, 

A  Power  is  the  product  of  two  or  more  equal  factors. 
The  degree  of  the  power  is  indicated  by  an  exponent. 

a?  =  a  a  a, 
and  a^  =  a  aaaa  a . 

Hence,     a^  X  a^  =  a  a  a  a  a  a  X  aaa 
=  a» 
a"  =  a  a  a  a  ....  to  n  factors, 
and  a*"  =  a  a  a  a  ....  to  m  factors. 

Multiplying  the  second  expression  by  the  first,  we  have, 

rt"*  X  a*  =  aaa to  m  factoi-s  X  a  a  a  ....  to  n  factors 

=  a  a  a  ....  to  (m  +  ?i)  factors 

=  a*"*"".     In  which  m  and  n  are  a?iy  numbers 


36  ELEMENTS   OF  ALGEBRA. 

whatever.  Similarly  for  the  product  of  more  than  two 
powers  of  a  factor.     Hence, 

The  i^owers  of  a  number  are  multiplied  hy  adding  the 
exponents. 

If  the  multiplicand  and  multiplier  consist  of  powers  of 
different  factors,  we  use  a  similar  process.     Thus, 

3m^  X  2  111^11?  X  5  m'^n^  =  o  X  2  X  5m 771  m m m  mmmnim 

X  n  n  n  n  n 

a"fe"*  X  a^lf  =aa  a  ....  to  n  factors  x  aa  a  ....  to  p  factors 

Xbbb to  m  factors  Xbbb  ....  to  r  factors 

=  aaa  ....  to  {n-\-p)  factors  x  bbb  ....  to  (m  +  r) 

factors 
=:  a""*"^  5'"  '^'■.     Hence,  in  general. 

To  Find  the  Product  of  Two  or  more  Monomials.     To  the 

product  of  the  numerical  coefficients  annex  the  factors,  each 
taken  with  an  exponent  equal  to  the  sum  of  the  exponents 
of  that  factor. 

Notes:  1.  When  no  exponent  is  written,  the  exponent  is  1.  Thus,  a  is 
the  same  as  ai,  &  as  fti. 

The  exponent  is  used  to  save  repetition. 

2.  We  read  a^,  a  square,  and  a*,  a  cube,  because  if  a  represents  the  number 
of  units  of  length  in  the  side  of  a  sqiiare,  and  the  edge  of  a  cube,  then  ffl2  and 
a3  will  represent  the  number  of  units  in  the  surface  and  volume  of  the  square 
and  cube,  respectively. 

Illustrations. 

11  mi9  X  10  m}^  =11  X  10  mi^  +  lo  =  no  ni^^. 

3a'^bcmX2ah'^cmX  5abc^m^  =  '3  X  2  X  5a2+i+i6i+2+ici+i+«mi+i+2 

=  30  «4  ^i  c*  m\ 
3a^fe*c3  X  4rt  X  6'cJ  =  3  X  4  a'  +  ift^  +  J  c3  +  ^  =  I2a^bc*^\ 
S'x^i/"  X  2^x-3  X  x^y""  =  25  +  ^x2-3+57/"  +  "  =  2 x*i/2". 


ALGEBRAIC   MULTIPLICATION.  37 

Exercise  13. 

Find  the  product  of : 

1.  0?  and  7  a?^ ;  3  a  a;  and  ^  c^t^  \  a? ha?  and  2  a^ W a?. 

2.  ^xyz^^  and  Is^t/^mn;  ^abcdm^7i^  and  ^a^lf^c^d^mn. 

3.  |^/2^.3yaud  |a6366y"  +  ';  3a3a:«3/7andf  ai«^a:8  3^*2. 

4.  3  a  a:^y^  and   10  a^^xy^^ ;  J  x"*  if  and  |  rr*  y^. 

6.    3aftca;^V  and  f  rt-^j^c^^^  \a'hh^xy  and  | a Z>iOc V^^"*. 

6.  f  a'-fc-af  /  and  .2  a^  h^  2^  f ,  a?  f  and  a^^i/S. 

7.  a'-iz-and  tt-6'";  a-^-^V^^and  5.7a:-i3/-i. 

8.  ahx]/^  and  a?-}?x^y\  a"'+'i>'*  +  '*  and  a'"-''/;"-'". 

9.  .55a;-'  +  '//-''  +  '  and    .5  a;''+«/-'';  .3  a2-"';j;8- n  ^j^^ 
a* 2:";  5a-H".r^  and   ^oa\hx'^\ 

10.  2«^2:,  a-?/,  «2//,  a^j^i/,  and    a  ^. 

11.  a*-,  fc",  3c',  a',  5',  c*",  and  c?'. 

12.  2c'"</,  «c",  r/22:",  a"*  a;*",  and  ci 

13.  ai  VI  x^,  a^  ni  x^,  a  m  x  y,  and   2  a^  71^  x^  if. 

14.  a^z^,  «^?/^,  ala;"i,  a^y~^,  5fa"la;^,  and  5^2:^yV 

15.  fa^ma:",  jTnta:*,  .Sarr"*.  5.1  w',  and  a-^'x-', 

16.  3y*,  a-^y'^z^,  (^ b"",  a^b',  f  «^.7/",  and  ^a^J"'^-. 

17.  (a  4-  ft),  5  («  4-  6)2,  3  («  +  6)^  ^  (^  +  &)^  and  (a  +  bf. 

18.  (a  +  6)  (c  +  (if,  {a  +  bf,  3  (c  +  df,  and  (a  +  6)7  (c  +  rf)2. 

19.  3  (a  +  6)-  (a:  -  y)-,  J  (a  +  6)«,  and  |  (a:  -  yf. 


38  ELEMENTS   OF  ALGEBRA. 

22.  Algebraic  Multiplication  is  the  operation  of  adding 
as  many  numbers,  each  equal  to  the  multiplicand,  as  there 
are  units  in  a  positive  multiplier ;  it  is  also  the  operation 
of  subtracting  as  many  numbers,  each  equal  to  the  multi- 
plicand, as  there  are  units  in  a  negative  multiplier.    Hence, 

The  multiplier  shows  that  the  multiplicand  is  taken  so 
many  times  to  he  oAded,  or  so  many  times  to  he  suhtracted. 

Thus, 
(+  6)  X  +  4  =  (+  6)  +  (+  6)  +  (+  6)  +  (-f  6)  =  +  (+  24)  =  -f  24 
(_  6)  X  +  4  =  (-  6)  +  (-  6)  +  (-  6)  +  (-  6)  =  +  (-  24)  =  -  24 
(+  6)  X  -  4  =  -  (+  6)  -  (4  6)  -  (+  6)  -  (+  6)  =  -  (+  24)  =  -  24 
(-6)  X-4=  -(-6)-(-6)-(-6)-(-6)  =  -(-24)=:-f  24. 

■  The  sign  of  the  multiphcand  (6)  shows  that  the  product  (24)  ii^  in 
the  positive  and  negative  series  of  numbers,  respectively;  and  the 
sign  of  the  multipl  ier  (4)  shows  that  the  first  two  products  are  to  be 
added  and  the  last  two  are  to  be  subtracted.     Hence, 

The  sign  of  the  multi-plicand  shows  what  series  of  numbers 
the  product  is  in,  and  the  sign  of  the  multiplier  shows  what 
is  to  he  done  ivith  the  product. 

Law  of  Signs.  The  product  of  two  factors  is  positive  when 
the  factors  have  like  signs,  and  negative  when  they  have  un- 
like signs. 

Since 

-2x-2  =  +4;  -2x-2x-3=:-h4X-^3==-12; 

-2x-2x-3x-4--12x-4  =  +  48; 

_2x-2x-3x-4x-5  =  +  48x-5  =  -240; 

etc.     Hence, 

The  product  of  an  even  number  of  negative  factors  is  posi- 
tive; of  an  odd  number,  negative. 


ALGEBRAIC  MULTIPLICATION. 

The  change  of  signs  may  be  illustrated  as  follows 


39 


>  + 


ax-Sr^ 

K. 

ox+y 

»^y»w.s 

ax-3 

V 

QX-^3 

VCATT 

ClX-2 

< 

ax+i 

Q  X-/ 

^ 

ax+/ 

^ 

A    * 

k 

-QX*Jl 

-ax+/ 

. 

0     , 

-ax- A 

-fly-3 

-CtX-i' 

^ 

7     ...     - 

-ffX+J^: 

•CTK^S^ 

k ■ 

-ax-s 

-| 

-< z^ 

Let  the  measuring  unit  be  represented  by  a. 

From  A  (o),  the  starting-point  on  the  scale,  measure  toward  the 
right  and  left.  The  products  of  +  a  and  —  a  by  the  factors  from  -f-  5 
to  —  5  are  : 

aX-l-5,  aX+4,  ax+3,  rtX4-2,  ax  +  1; 
a  X  -  1,  a  X  -  2,  a  X  -  3,  rt  X  -  4,  a  X  -  5  ; 
-aX+5,  -rtX4-4,  -ax+3,  -ax+2,  -aX  +  1; 
-aX-1,  -aX-2,  -aX-3,  -ax- 4,  -aX-5; 

respectively.     The  directions  taken  by  the  products  are  shown  in  the 

figure. 

ninstrationB. 

x«y»  X  -x*z  X  -^j/z^  X  -^xz^  X  -  4  yz^  =  -f-  f  X  ^  X  4x«j/*2»o 

X^y-*  X  -  fx-y-Z  X   -  yz-'-  X  -  X~^'*  =    ~  ajm  +  n-Sny-n  +  n+ljl-r 

Exercise  14. 
Find  the  product  of : 

1.  5  a,  —  3  /),  7  c,  —  2  a*,  —  11  a^,  and  a;  a^x,  —ay^, 
a  a^,  and  —  xy. 

2.  ahx,  —ay^,  —a  X,  and  a^a?\  —al^,  —hc^,  —cd?, 
—  a,  —  a^,  —  a^,  and  —  5  a* 


40  ELEMENTS  OF   ALGEBRA. 

3.  —  a,    he,    —  1,    ^,  1^  a^,   ^x y,  and    75  a;    ax,  ex, 

—  m  X,  —  2/**,  and  .3  ?/i. 

4.  ^abc,  —d,  ax,  —1,  and  ^axyz;  a'^af,  af^y*",  a'^V, 
and  a  b. 

5.  —a^x,  3x,  ah^,  ay,  az,  and  axyvw;  axy,  —^a^V, 
and  -  SJa^&'a:"*?/". 

6.  -  a'^hc,  2  h'^cd^  -  .5  a^ccl^,  -  f^  a-^H-^^c^^d'^^, 
and  a  h^  d*. 

7.  -1,  a- 3,  a^7^  a  0^-5,  aio^-3^  a-H-^a^,  and  -Mr/2 

8.  aaP,  —  a\  —  1,  .3  a  x,  and  —  a^^/^;  ^^^.^  _  ^|^  ^|, 
and  —  a^  a^i. 

9.  — my,  mx,  — mn,  — xy,  and   it* 3/3;    3  aa  Jt   and 

—  .7  «i  Z>i 

10.  a",  a^",  a^",  a^**,  and  a^".     Express  the  result  in 
two  ways. 

11.  2^ij-'x-^,  mx'^y'^',  -3"  ^"2:-!,  and  -2-^po(^f. 

12.  32",    -23«  X  3'^^^    32«,  -  23"  X  3*«,    S^"  x  2«,   and 

—  26"  X  3«  +  i. 

23.    Example.     Multiply  a  +  &  by  m  ;  also  a  —  fe  by  m. 

The  symbol  (a  +  h)  m  means  that  m  is  to  be  taken  (a  +  V)  times. 
Hence, 

Process. 

(a  +  h)m  —  m -\- m  ■\- m  + taken  a  +  h  times 

=  {m+m+m+  —  taken  a  times)  4-(7w  +  m  +  m+  —  takenft 

times) 
~  am  -\-l)m.  (1) 


ALGEBRAIC  MULTIPLICATION.  4l 

Also, 
(a  —  h)m  =  m-\-mi-in-h  ....  taken  a—b  times 

=  (m  +  7»  +  m-f- taken  a  times)  —  (m  +  m+m+....  taken  K 

times) 
=:(rnX  a)-(mX  b) 
—  am  —  bm.  (2) 

Similarly,  (a  +  ft  —  c)  m  =  a  m  -f-  6  m  —  c  m. 
These  results  are  obtained  by  multiplying  each  term  of  the  multi- 
plicand separately  by  the  mnltiplier.     Hence,  in  general, 

To  Multiply  a  Polynomial  by  a  Monomial.  M%dtiply  each 
term  of  (lie  muUiplicaiul  by  Ute  multiplier ^  and  add  the 
resiUts. 

Exercise  15. 

Multiply : 

1.  hc-^-ac-ab  hy  abc;  S aH^ - ^  hh^ -  ^ c^  hy  -f^a^b^^. 

2.  5a^-b'-2c^  by  a^b^c^^;  .6  3^- .5  2^y^- .32^y^ 

-  .2a^  by  .2x^f. 

3.  j  W.2  —  ^mii  -{-  ^n^  by  ^mn;  x  —  y  —  ^x^y^  by  xy. 

4.  fa-^j&2_^^^a62  by|«Z^2.   a' - a^lr^-ab  hy  ah^. 

5.  6a2a3-  .5a^b^x^^  -\-  .2h^2^  by  ^  ab  3^;  pxT-qx* 

—  r  by  p2^  r. 

6.  3a'"-»-2  6"-H4a'"6"  by  a&2;  .4a— "  5'''-Ja-*'6' 
+  ft3^  by  I  a-^-^'ft'. 

7.  a*--3a'"i—4-&'*  by  a'-J'^";  2^2:^  -  2iyi  +  2^a;iyi 
by  2^x^yi 

8.  a?-a2jf-an  +  5iby  aU^;  a;*-2rty^4-a:^i^t-.6y* 
by  xhj~^. 


42  ELEMENTS   OF  ALGEBRA. 

Find  the  product  of : 

9.  x^y^-4:  x^f+  4/,  xh/,  and  -2i/;  m''^  -  2m^^7V'' 
+  n^\  m~^,  n-\  and  in"  n\ 

10.  ^^26-4  +  1^6-32:  + 1 62 2:2^  laV^,  %b-^x,  and  ^aH^a^. 

11.  a;3  —  7/F    a;3,  2/3^  and  —x'^i/^;  a—M,  a?,  63,  a^b^, 

—  a2  jf    and  —  a  b^. 

12.  x^—i/,   x^,  x^y^,  —x^yi,  |  ?/f ,  J  2:t,  —  |- ?y^,  and 

.21       .2_1_ 

rr^  7/4. 

13.  J  -  .2  6f  :2:2  +  ;3  7;  -^i  _  ^1^  J  ^1^  _  il  ^2^  and  J  b^  xi 

14.  1^--^'"      -y  «--6-"'  + |6,    'Sa-"\    -rjb-'^,    and 

15.  «"+«  +  f/"6'«  +  (fc"»6'*  +  6'"+",  a.'",  ft"*,  rr",  6"",    and 

24.    Example  1.  Multiply  m-{-ti  by  a;+y;  also  m^n  hyx  —  y. 
(m  +  w)  (v+y)  means  that  a:  +  ?/  is  to  be  taken  m  +  n  times.     Ifence, 

Process 

(m  +  n)  X  0-  -T-y)  =  (^'  +  y)  +  (^'  +  2/)  +  C-^'  +  y)  4-  •  • .  •  taken  w  +  n  times 

=  [(*'  +  ?/)  +  (•^■  +  ?/)  +  (x  +  y)  +    ...  taken  m  times] 

+  [  (*■  +  ?/)  +  C^'  +  y)  +  (^  +  y)  + taken  n  times] 

=  (x  -{-  y)  m  +  (.r  +  ?/)  n 

=  (1)  Art.  23,  ??*  a-  4  m  y  +  n  x  -\-  n  y.  (1) 

Also, 
(m  +  n) (x  -  y)  =  (x-y)  -{■  (r~y)  +  (x-y)-\-  ....  taken  w  +  n  times 
-■  [(-^  —  ?/)  +  (t  ~  ?/)  4-  (?'  —  ?/)  4-  —  taken  m  times] 
4-  [(^  "  y)  4-  (j"  —  ?/)  4-  (r  —  1/)  -f  —  taken  n  times] 
=  (x  ~  y)  m-{-  (x  -  y)  n 
—  (2)  Art.  23,  nix  —  my  -\-  71  x  —  n  y.  (2) 

Similarly,  (7n  -\-  n  +  p)  (x-\-  y  —  z)  =  m  x  +  7n  y  —  mz  +  nx  +  n  y 

—  71  z  -\-  p  X  -j-  p  y  —  J)  z 


ALGEBRAIC  MULTIPLICATION.  43 

These  results  are  obtained  by  multiplying  each  term  ot  the  multi- 
plicand separately  by  each  term  of  the  multiplier,  and  connecting  the 
products  with  their  proper  signs. 

Example  2.     Multiply  j*-i*+2x^-x-5  by  x*-h3x*-\-  5. 

Process. 

x«-     r«-j-2x«-a:-5 
X*   +  3  x«  +  5 


x"  -     x«  -H  2  x«  -     x«  -  5  X* 

+  3z»  +6x*-3ir*~3x8-  i5x« 

+  5  x«  -  T)  j;5  +  10  x*-*  -  5  x  -  25 

xW  -H  2  x»  -I-  7  x«  -  «  X*  -  3  x8  -  16  x»  +  10  x2  -  6  X  -  25 

Explanation.  Multiplying  each  term  of  the  multiplicand  by 
each  term  of  the  multiplier  and  connecting  these  results  with  their 
proper  signs,  we  have  x**'  —  r*  -f-  2  x*  —  x*  —  5  x*  +  3  x*  —  3  x*  4-  6  x* 

-  3  X*  -  15  x«  +  5  x«  -  5  x«  +  10  x2  -  5  X  -  25.  Umling  like  terms, 
for  a  simplified  product,  we  have  x^<>  -f-  2  x*  —  3  x^  -f  7  x*  —  8  x*  —  15  x* 
+  10  x«  -  5  X  -  25. 

The  process  used  in  practice  is  shown  above.  The  first  line  under 
the  multiplier  contains  the  product  of  the  multiplicand  and  x*.  The 
second  contains  the  product  of  the  multiplicand  and  3  x*.  Etc.  To 
facilitate  adding,  write  the  several  products  so  that  like  terms  shall 
stand  in  the  same  column. 

Hote.  It  i^  convenient  to  arrange  the  terms  of  the  multiplicand  and  multi- 
plier according  to  powers  of  some  common  letter,  ascending  or  descending. 

Example  3      Multiply  f  a  x  +  f  x^  +  ^  a*  by  f  fl^  +  §  x^  -  f  a  x. 

Solution.  Arrange  the  expressions  according  to  the  descending 
powers  of  r.  Taking  the  multiplican<l  |  x*  times,  we  have  x*  +  a  x' 
-f  ^  rt^x^.  Taknig  it  —  |  n  r  times,  and  writing  the  proiluct  so  that 
like  terms  nhnll  stand  in  the  same  column,  we  have  —  ax*  — a*x* 

-  1^a*T,  Again,  Uiking  it  |  a^  times,  and  writing  the  pnnluct  as  be- 
fore, we  have  ^a^  x^  -h  ij^a*  r  +  \  a*.  Adding  the  partial  products, 
we  have  x*  -I-  ^  a*,  or  arranging  alphal>etically,  {  a*  -f-  x* 


44  ELEMENTS  OF  ALGEBRA. 

Process, 

f  x2  -  f  a  a:  +  f  a^ 


—    ax^  —     a'^  x"^  ~  ^  a^  X 

+  ia^x^  +  la^x  +  ia* 

Example  4.  Multiply  -  Sx^'  +  ^y^  -  .Sx^^^^  y''  +  ^  +  3.3j:"'^  +  ^ 
by  —  .2  j:™«/«-2  +  4  a;'"-' i/«-^ 

Process. 

3.3  a;*"*/" +  2  —    .3    x"" +  ^  y'' +^  —3  x"' ^'^y'' 

4     x™  - 1  ^'»  -  ^    —    .2    a;"»  ,y"  -  ^ 
13.2r*"»-i/«+i  -1.2    a:-'"**/-'"  -  12.00 x^^  +  'j/an-i 

-    .66a;2'«?/2»,_^       06a;2"»  +  i  j,2«-i^  g^zm+a^^n-g 

13.2  j2m-1^2n  +  l_l  86^:2 '"y^»-  11.94x2"'  +  !  ^2«-I_^    (5  a,2m  +  2^2«-2 

Explanation.  Arrange  according  to  the  ascending  powers  of  x,  as 
shown.  The  product  of  the  multiplicand  by  4x'"-'  ^"  ~  ^  gives  the 
first  partial  product,  as  shown  on  the  first  line  under  the  multiplier. 
The  product  of  the  multiplicand  by  — .2x'"^"-2  gives  the  second 
partial  product.  Taking  the  sum  of  the  partial  products,  we  have 
the  product  required.     Hence,  in  general, 

To  find  the  Product  of  two  Polynomials.  Multiply  the 
multiplicand  hy  each  term  of  the  multijplier,  and  add  the 
partial  products. 

Exercise  16. 

Arrange  the  terms  according  to  the  powers  of  some 
common  letter,  and  multiply : 

1.  r/2  J^h'^-ah  by  «  6  +  ^2  +  ^2  .  a'^-2ax  ^  4.x^ 
by  «2  +  4  .^2  +  2  a  a^. 

2.  x^  -\-  y^  —  x^  y"^  by  x'^-\-y'^\  x  +  y  -{-  x  —  y  by  x-^-y 
-x  +  y. 


ALGEBRAIC   MULTIPLICATION.  45 

3.    y— 3  +  /y2  by  2/-9+2/2;  a^tj  —  azi-ij^  —  a^  by  y-^  a. 

^   J  a;2  -  I  X-  -  f  by  J  a;2  ^  I  ^  __  1 .  1  6  «2  4.  1  2  a  6 
+  9&2  by  Aa-.:n, 

5.   x^—  i/-\-  X  —  fj  by  x^-^  y^-\-  x  —  y  \  ^s^  —  ax  —  ^a^ 
by  \  x^  —  ^ax  -\-  ^a^, 

+  \d^  by  2  a;^  +  «  2:  —  ;|  ^3. 

7.  n7^-Dx'^-x^^-2^-x  +  2hyx^-2x-2, 

8.  3a2-2a3_2a  +  l  +  a*  by  3ft2+ 2^3+ 2a  +  l  + ^*. 

9.  1.5  2:8  +  1.5  2^2  +  .5  a:*  +  .5  2:  +  2:^  +  1  by  a^  -  .5  a; 
+  1  +  a:*  -  .5  2:8 

10.  1  +  9  a  +  5  a3  +  3  a*  +  7  a2  4-  a^  by  4  a2  -  3  a8 
4-  a*  4-  4  -  4  a. 

11.  4  2:2^24.  82:^3+16/  +  2a:3y  _^^  by  2:  -  2  y. 

12.  2^12-  a:3^6  4.   2^6^-  2:87/24.   y8   by  y«-|-   2:8.   242^2 

—  3^  y  —  X  i^  -\-  x^  -\-  y^  by  x  -\-  y. 

13.  rt2  _^  ^2  _^  ^  _  ^  5  _  rt  c  _  ^  c  by  rt  +  &  +  c. 

14.  ^2  _^  ^,2  _^  ,.2  4.  J  c  +  a  r  -  a  6  by  a  +  fe  -  c. 

15.  a6  +  crf4-ac4-6c^  by  ab-hcd^ac  —  bd. 

16.  i2  4.  y2  _  3  3^  _  y2  by  2  a:  +  2  3^  -  2  (2:  -  y). 

17.  3  (m  +  n)  —  .1  X  (a  +  h)  by  a  -  b  -^  .1  (m  ^  n). 

18.  a2f^-\-bx''-\-r  by  a2:*  +  62;"+r;  2;^  +  yi  by  2:^  —  y^- 

19.  a-  +  6"  by  a'"  +  b";  oT  4-  6*  by  a"  -  6*;  2:2  _,_  j  by 
a:i  +  bl 


46  ELEMENTS   OF  ALGEBRA. 

20.  Sx"^-^  -  2/-'  by  2x-  S'f;  ax'''  +  &^"+  ahx 
by  a  x^  —  bx^  —  1. 

21.  'Sa^^'x+'Sa^y  +  a'''  by  a'"- a" +  2:;  x^-y-i  by 

2^2  —  y. 

22.  .2ai-.3&t  hy.2ai+.3bh^xi  +  xUji-hy^  hy  x^-yk 

23.  a;^  ?/~t  +  y~^  +  x^y"^  +  a;^  by  x^  —  y~i. 
Find  the  product  of: 

24.  1  +  2;,  1  +  2;^  and  1  +  x^  —  x  —  a^. 

25.  a;  —  2  a,  X  —  a,  x  +  a,  and  a:  +  2  a. 

26.  3  2;  +  2,  2  a:  -  3,  5  :z:  -  4,  and  4  a;  -  5. 

27.  ^2  —  :r  +  1,  a;'-^  +  :r  4-  1,  and  x'^  —  x'^  -{-  1. 

28.  rr^  —  a  x-{-  a?,  x^  +  a  x  ■}-  a?',  and  x^  —  a'^  a;^  +  a*. 

29.  «  +  6,  a  -  6,  3  a  +  &,  and  a^ -2o?h  -  a}?'  \  b^ 

30.  rr  +  &^  «"*-&",  ft''"+a"*6"+?)'",  and  a2'«_,^«^H  +  ^a«^ 

25.    A  Binomial  is  a  compound  expression  of  two  terms ; 

SiS,  a  —  b;  ab  +  2b\ 

In  each  of  the  following  products,  observe  that : 

2a;  +    3  2  a:  +    3 

2x  +    5  2a;   -    5 


4a;2+    6  a;        -~  4x2+    ^^^ 

10  a; +15  -10  a; -15 


4x2 +16  a; +15  4^2-    4  a; -15 

2a;-3  2  a;   -3 

2x   +    5  2a;  —    5 


4a;2-    6  a;  4^2-    6  a; 

10  a; -15  -lOx+15 


4x2+    4  a; -15  4x2-16x+15 


ALGEBRAIC    MULTIPLICATION.  47 

I.  Thejirst  term  is  the  common  algebmic  term  of  the  binomials 
multiplied  by  itself,  or  the  square  of  the  common  algebraic  term. 

II.  The  second  term  is  the  al«;ebraic  sum  of  the  other  two  terms  of 
the  binomial  expretssious  multiplied  by  the  common  algebraic  term. 

III.  The  last  term  is  the  algebraic  product  of  the  terms  which  are 
not  common  to  the  binomial  expressions.     Hence, 

To  find  the  Product  of  two  Binomials,  having  one  Common 
Algebraic  Term  yidd  toy  ether  the  nfpiare  of  the  common 
tertn,  the  abjebraic  aum  of  the  other  two  tervis  multiplied  by 
the  cmiimaii  terniy  aiul  the  algebraic  proditct  of  the  terms 
which  are  iwt  common. 

In  general,  {x  -\-  a)  (x  ±b)  —  2^  +  {a  ±b)  x  ±  ab  (1) 

(x-a){x±b)  =  x^-\-{-a±b)x^^ab       (2) 

In  which  a,  b,  and  x  represent  any  numbers. 

Hotel:  1.  It  is  of  the  utmost  importance  that  tlie  student  sliould  learn  to 
write  tlie  products  of  binomial  expressions  rapi<lly,  by  inspection. 

2.  To  square  a  monomial,  multiply  the  numerical  coefficient  by  itself ^  and 
multiply  the  ejptment  of  eacli  letter  by  two.  The  proof  is  evident.  Thus,  the 
square  of  2  a*  6»  =-  2  X2  a*  ^  *  A*  >< «  -  4  ofta-. 

Also,  (36-"ar«)a  =  3  X  36-»x2a*«  ^a  -  96-2««a;2m. 

Examples.     Write  the  product  of  the  following  by  inspection  : 
(2  a;  +  7  y)  (2  I  -  5  i^);  (a  -  9  6)  (a  -  8  6) ;  (a  -  6)  (a  +  1 3) . 

Solution.  Squaring  the  common  term,  we  have  4x^.  Taking 
the  algebraic  sum  of  the  other  two  tenns,  +  7  y  and  —  5  y,  we  have 
+  2y.  Multiplying  this  sum  by  2  a;,  we  have  +  4a:y.  Taking  the 
algebraic  product  of  the  terms  not  common,  +  7  y  and  —  5  y,  we  have 
—  35  y*.     We  thus  obtain  4x^  -i-  4  xy— 3b  y^  for  the  product. 

Similarly,  (a-96)  (aSb)  =  a«+  (-96-86)  X  rt  +  (-96)  X  (-86) 

=  a3-17a6-|-72  6^. 
Also,         (a-6)(a+13)  =  a2+(-6  +  13)  X  a-f  (-6)  X  (4-13) 

=  a2  +  7a-78. 


48  ELEMENTS  OF  ALGEBRA. 

Exercise  17. 

Write,  by  inspection,  the  products  of  the  following : 

1.  (a -3)  (a +  5);    (6+6)(&-5);    {x  +  4)  {x  +  S) 
{x  -  4)  (0^  +  1)  ;  (^  -  7)  (x  +  2). 

2.  (x  -  8)  (^  -  6) ;    (a  +  9)  (a  -  5);    {a-  8)  (a  +  4) 
(2x-4:)  (2  a^  -  5)  ;  (3  ^'  +  7)  (3  ^  -  5). 

3.  (0^3-37/2) (^3_ 4 2^2).  {x-7y)(x  +  Sy);{a"^-l){a-+2) 

(3a:5-5)(3a:^-4). 

4.  (2  a2  2/3  +  4)  (2  ^2  f  _  8)  ;     (3  a  a;  -  4)  (3  a  2:  +  7) 
(a:3  +  3  a)  (a;3  -  4a;)  ;  (ai^  -  3  a2)  (a;^  +  2  a^). 

5.  (2a;  +  a)(2a;-2a);  (2:c"+ 5a)(22;"-3«);  {Sx-2y) 
(S  X  +  y)  ;  {-  6  m  +  2  2^)  {4.m  +  2  x^). 

6.  (:r-a)(2:-5a);  {a-5b)(a  +  Sb);  {a^-2x){a^-6x); 
(5xio+3a2)(5:z;io_4a2). 

7.  (32/2-5a:^)(2  2/2-5:?;5);  (3  a^  +  2ab)(3a^-4:a¥); 
(a"  +  3)  (a»  -  b). 

8.  (4  a  +  6)  (4  a  -  c) ;  (2  &  -  5  a)  (2  c  -  5  a) ;  (a  y  4-  i^;) 
(a.^  +  i^);  (af-l)(al+|). 

9.  (2  2:^  +  1)  (2  xi  4-  12)  ;  (2  a^  -  3  ax)  (2  a^  +  b)  ; 
(x-  .Sx^y''){y-  .Sx^y"*). 

26.  (:r4-7/)(2;-  y)  =  x'^  +  {y  -  y)  X  x -]- (i- y)  X  (-y) 
=  2^2  —  ?/2_  Xn  which  a;  and  y  represent  any  two  numbers. 
Hence,  in  general, 

To  find  the  Product  of  the  Sum  and  Difference  of  two 
Numbers.     Take  the  difference  of  their  squares. 


ALGEBRAIC   MULTIPLICATION.  49 

Examples.  Find  the  product  of  (2  a"*  -f-  3  b-")  (2  a"»  -  3  &-*) ; 
(8;)*  4-  ll2*)(8;>*-  Hz*). 

Solution.  (2a«  +  36-")  X  (2a"  -  3 ft-*)  is  the  square  of 
2  a*",  or  4  a*  •",  minus  the  square  of  3  6"",  or  9  6-**.  Therefore, 
(2  a"»  +  3  6-*)  (2  a"  -  3  6-*)  =  4  o* »»  -  9  6-  «*. 

Similarly,  (8/)*  4-112*)  (8p*  -  11  z*)  =  64/)«  -  121  z. 


Exercise  18. 

Write  by  inspection  the  product  of  the  following : 

1.  (2x-^'Sy)(2x^Sy);  (x -{- 2ij){x-2y);  (5  +  3  a;) 
(5-3a:);  (5a:+  11)  (5  2; -11). 

2.  {2x-h  l)(2a;-l);   (2x+  5){2x-b);  (5xy  +  3) 
(5  a;  y  —  3) ;  (c  -f  a)  (c  —  a). 

3.  (c2  +  a2)  (c2  -  a2) ;    (m  n  -H  1)  (wi  n  -  1)  ;  («  y^  4.  j) 
{af --}))■  (a2r2+  l)(a2a:2_  1) 

4    (a:*  +  7/)  (ar*  -  /);    (1  -  pq)  (1  -f-  pq)-    {m  -  n) 
(m^-n)\  (a"* -fa")  (a* -a"). 

5.    {bxr^^^y^{oxy-^-V4.f);  {h^+^f)(^2?-?>f)) 
{2!^-Zx){j^+  3a;). 

G.    (2aa:4-  fey)(2aa;-6y);  (m"' +  7i-»)  (m"'- 7^-»); 
(10  a-"  -  13  6—)  (10  a—  +  13  6—). 

7.  (mi  +  rA)  (mi  -  ni)  ;  (4  ai-  20a;io)  (4ai  +  20  a:iO)  ; 
(ai-6-f)(«i  +  6-i). 

8.  (11  a:i  +  30  ?/*)  (11  a:*  -  30  y*)  ;  (15  a2  6^  -  16  a*  6^) 

9.  (i«6-2+56-Ja:-i)(Ja6-2-i6-ia;-i);  (a+6)(a-6) 


50  ELEMENTS   OF  ALGEBRA. 

10.  (ah+l)(ab~l)(aH^-\-l);  (2a'"+ 4a")(2  a"'-4a") 
(4^2"^+  16^2"). 

11.  (5 a^  +  6&2)  (5 a^  -  6h^)  (25d^  +  366*) ;  {a-^  +  aH^) 

12.  (rc-l  +  x-Uj)     (x-^  -  x-'y}    (^x-^  +  x'' f}  ; 

(f  cr-  +  i  If)  (f  c-  -  |-6«)  (f f  c-'^-  +  }f  62"). 

Queries.  In  finding  the  product  of  monomials,  why  add  expo- 
nents of  like  factors  ?  What  is  the  product  of  a^  and  a^  ?  Prove  it. 
Why  is  the  product  of  an  even  number  of  negative  factors  positive  ? 
How  prove  (1)  and  (2)  Art.  25  % 

Miscellaneous  Exercise  19. 

Multiply : 

1.  2  ^2"  -  a"  +  3  by  2  a2-  +  a"  -  3  ;  5  +  2  a;2«  +  3^ 
by4^«-3^2a 

2.  ft^  +  2  a^"  -  3  by  5  -  J  a"  +  2  ^2*  ;  J  a;i  -  5  +  8  a;t 
by  \x^  +  lx~'^. 

3.  3  ^1  _  a  -  a^  by  f  a^  +  a"!  -  6  a-i ;  2  a*""^  &-"' 
+  a-'^  6^  by  3  a'"  ?^'^  -  ««^  Ir^K 

4.  ^x''if—'ix-''y-^  by  4  ^«?/ +  5^2a^26.  ^t"  +  «-t" 
by  ai"  +  «~^". 

5.  .3  ft*  -  .02  ft36  +  1.3  «2^2  +  .5  a2,3_  1  2  h^  by  .3^2 
-  .5  a  &  -  .6  2>2. 

6.  1  -  2  ^^  -  2  ^i  by  1  —  .T6  ;  al  -  8  «-t  +  4  a-^  -  a^ 
by  ia~^  -\-  a  -\-  I  a-\ 

7.  2x^-x^-3x-^  by  2  x'^  -  3  x~^  -  x-^  ;  a"  -  1 
+  ft- "  by  a^  +  ft~i 


ALGEBRAIC   MULTll'LICATlON.  51 

8.    ^■""■'■^  -  x-"  +  '  -  X-+  x"-'  by  0;"  +  '^  -  jJ"  -  ^;  +  1  ; 
a,'»+3a;"-*-2a,"-^  by  2  .x'  +  ' +  it:-  +  »  -  3  x*. 

lU.   3^- 2a;'"  +  '  -  5a;'"  +  =»+af*  +  ^  by  3a;"-3  +  2  3;"-* 


11.  a:"  +  ^-  3x-+*+  x*+''-  2ic-  +  *  by  2a;'-"+  Sic^'  — 

12.  5a;"-V-*-'-2a;--^/-^'-u;**-'/+'  by  3  a;*+ */"' 

13.  //«.'  +  '  — 3m'*?i+m'-^7i2—?/i'-^t^  by  m*"-^— 3m-?i 

14.  2.«;"+V"'4-3a=*'"'/'"^-a;"+Y'"'  +  4x«+V'"' 
by  2./;■+y-'**'-3^-•^Y-''  +  .tV"''  +  4x"-y-^ 

15.  x-+*y-*  +  x'-^*if-'  -  2a;-  +  V~"  -  4a:-  +  'y-* 
+  4a;»— y^"- 

16.  (y+  a;-)(y- .»:-'") ;  (i^^r -f •^"VXi^rHf^c-/^). 

17.  (x''  +  y~)ra:'*-y-);  (a:i  -  5)  (xi  ^  4),  (7  x  ^  3y-^) 
(7x+3rO. 

18.  (4  X'  -  5  x-^)  (4  a;  +  3  x-^) ;    (f  A  b'^  -  ^^  a^I)^) 
(|cU-t  +  ^a^65);  (a- 4-  7  +  3a-'')(a~  +  7-3a-''). 


52  ELEMENTS  OF  ALGEBRA. 


CHAPTEE  V. 
INVOLUTION. 

27.  Involution  is  the  operation  of  raising  an  expression 
to  any  required  power. 

hivokition  may  always  be  eflected  by  taking  the  expres- 
sion, as  a  factor,  a  number  of  times  equal  to  the  exponent 
of  the  required  power. 

It  is  evident  from  the  law  of  signs  that  even  powers  of 
any  number  are  positive  ;  and  Ihat  odd  powers  of  a  number 
have  the  same  sign  as  the  number  itself.     Thus, 

(—  m*  uY  =  (—  m*  u)  X  (—  m^  n) 
(-  m*  7i8)8  =  (-  m*?i8)  X  (-  w*  n^)  X  (-  m*  n^) 

=  -m< +4  +  4^8  +  8-1-8  =:-ml2u». 

(-  3  m8  ny  =  (-  3  m^  n)  X  (-  3  m^  v)  X  (-  3  w^n)  X  (-  3  m^  w) 

^ +  31  + 1  +  1  +  1^8  +  3  +  3+3^1  +  1  + 1+1  =  4.  81  mi'-^n*. 

(a**  6<^)~  =  a*^  b"  X  oT^  ¥  X  a"^  b*"  X ton  factors 

=  (a*"  X  a"*  X  a"*  X  ....  to  n  factors)  X  (6^^  X  6*^  X  6*^  X  ... . 
to  n  factors) 

/'Qm+m  +  m+.. ..  ton  terni8\   y^    /'^c  +  c  +  c+....  ton  termsA 

—  qM  X  n  ^  ^c  x  n 

—  Qmn  ^cn^  where  c,  m,  and  »  are  positive  integers f  a  and  6 
may  be  integral  or  fractional,  positive  or  negative. 

Similarly,  (a"*6<^# joO"  =  o*«"6<^"cZ*«  ....  ;?♦•".      Hence,  in 

general. 


INVOLUTION.  63 

To  Baise  a  Monomial  to  any  Power.  Multiply  the  exponent 
of  each  factor  by  tlie  exponent  of  the  required  power y  and  take 
the  product  of  the  resulting  factors.  Give  to  every  even 
poiver  the  positive  siyyi,  ami  to  every  odd  power  the  sign  of 
the  monomial  itsdf. 

Notes :  1.  Since,  aw  -  -  1  X  a"»,  the  nth  power  of  -  a™  -»  (-  1  X  a"*)" 
—  (—  1)*  X  «"•».  Or  we  may  write  ±  a'"»»,  for  the  nth  power  of  -  a»»»,  where 
the  positive  or  negative  sign  is  to  be  prefixed,  depeudiug  upon  the  value  of  n 
whether  an  even  or  odd  integer,  in  being  positive  and  integral. 

2.  Any  power  of  a  fraction  is  found  by  taking  the  required  power  of  each  of 

Uliistrations. 
(-3x'»/)«  =  -3''<«a:«'<V''         =-27x«y« 

Exercise  20. 
Write  the  results  of  the  following : 

1.  (4aH*)2;   (3a668)3;  (2  a; V)^  (^a^V^c^f;  (.1  a'»6-)^ 

2.  {1  a^V^f)  (llaJ2c8(fiO)2.  (-3cx3/2«)8;  {ZaH'^iff; 
{5abcc^y^z^y. 

3.  (-2a2)c2^7/*)8;  (-abcdxy^-  (-a^l^cf;  (-^^O*; 
{SaPc^;  {-2aV^f. 

4.  (lXa<68c2^)n;  (-2a2"6"»)^  {-Zxyzf\  (x^^^^^ "-)•"; 

5.  (-2a''66«.)8.  (^•)«.   (a«)a.    (ft ft -ic- 2)6;    (m"7i— )•"•; 

(-2)8;  (-«)«";  (-ir. 


54  ELEMENTS  OF  ALGEBRA. 

6.  n(2  a  b'^  c  n "i)*;        n  {n^  m^*"")";        m  (m"  a"^)"; 

7.  2  a(-  3m7i3a;*)3;  m(-  3  aiojs^^e^*)*;  a>^  (3  a-^J-^; 
a(a«-')«. 

8.  C2x^y\zhy\    (-ra;V"^T;     (-3ain-"c)"; 
(_  3«-''6't^iyA)6. 

Affect  the  following  with  the  exponent  7 ;  that  is,  raise 
each  to  the  7th  power. 

10.  {-x''y'^f-{ahh^f;{^^a})xy\  [{-x'^yf ',  {-T^mTf. 
Write  the  nth.  powers  of: 

11.  mia-'^ciyix-yy;  (a-3 d)^ '' (x-yf ;  3 (a-b-\-c-\-d) 

{a  —  xy. 

12.  a&c(a-6T(^  +  2/  +  ^T;  a''{x-y-^zf''\x-y'^f''\ 

28.    It  may  be  shown  by  actual  multiplication  that : 
{a  +  hy  =a-H6H2a&; 

(a-hy  =a^+h^-2ab; 

(a  +  b-\-cy       =a2-}-62_,_c2+2a6  +  2ac  +  26c; 
(a-b-cy       =a^+b-^-\-c^-2ab-2ac  +  2bc; 

{a-\-b  +  c-\-dy=a^+b^+c^-\-d^+2ab-]-2ac  +  2ad  +  2bc+2bd-\-2cd; 
etc.  etc.  etc. 

In  each  of  the  above  products,  observe  that  the  square  consists  of : 

I.  The  sum  of  the  squares  of  the  several  terms  of  the  given 
expression. 

II.  Tvnce  the  algebraic  product  of  the  several  terms  taken 
two  and  two. 


INVOLUTION,  55 

These  laws  hold  good  for  the  square  of  all  expressions, 
whatever  be  the  number  of  terms.     Hence,  in  general. 

To  Square  any  Polynomial  Add  together  the  squares  of  the 
several  tertns  aiid  twice  the  algebraic  prodtict  of  every  tvH) 
terms. 

Example  1.     Square  3  a»  —  4  x*. 

Solution.  The  squares  of  the  terms  are  9  a'  and  16  x^®.  Twice 
the  algebraic  product  of  the  terms  is  —  24  a^  x*. 

Therefore,  (3  a»  -  4  x»)«  =  9  a«  +  16  x"  -  24  a«x». 
Example  2.    Square  2  x'  -  3  x*  -  1. 

Solution.  The  squares  of  the  terms  are  4  x*,  9  x*,  and  1.  Twice 
the  algebraic  product  of  the  first  term  and  each  of  the  other  two 
terms  gives  the  products  —  12  x*  and  —  4  x*.  Twice  the  product  of 
the  second  and  third  terms  is  6  x^. 

Therefore,  (2x»-3x2-  l)a  =  4x»+  9x* -I-  1  -  12x»-4x«  +  6x«. 

niuatrations. 
(2  a"»  -  3  x-»)«  =  (2a  "•)2  -|-  (-  3  x— )«+  2  (2  a"»)  X  (-  3  x^") 
=  4  a*"  4-  9  r-«"  -  12  a"»  x"*. 

(x-V-Ky-*+|y'-iy)'=(^V)"+(-iar-tr*)*+(|y^+(-iyy 

+  2(x-V)  X  (-  ix"y-«)  +  2(r-«y'') 
X(§y'H-2(x-V)X(-iy)  +  2(-ix"y-«) 
X(|y«)  +  2(-ix-ir*)X(-iy)+2(fy») 
x(-Jy) 

+  |x-«y*+t-|r-«j/*+»-§x"y  +  ix*y-'» 


56  ELEMENTS  OF  ALGEBRA. 

Exercise  21. 

Square,  by  inspection,  the  following : 

1.  x  + 2;  m  + 5;  n-\-7;  a —  10;  2x -\- Sy;  a  +  Sb; 
a  —  Sb;  2x  —  Sy. 

2.  X  +  5y;  3x  —  5y;  2a  +  ab;  5x  —  Sxy;  5abc  —  c; 
xy-^-2y^;  a™  +  3  6"". 

S.2x  +  Sa^;     xy  +  x'^;     3  a-2 -f  5  a^-^;    1-x; 
1  —  cy;  m  —  1;  ab"^  —  I  ;  -J  a"—  .05. 

4.  1  a  6-2  + |6-ia;-i;      fff-r^'^      |  a-«  _  2  j-»«; 
ic-f 3/-f  +  J;    .0002a;'"  +  .005/. 

5.  I  m^  n^p^  —  ^mrf;  xy  +  yz-hxz;  2x^  ■]-  Sx  —  1; 
x^-2x+l;  x^  +  2x-4. 

6.  2a^—x  +  S;  a^—5x—2;  x^—2xy  +  y^;  4:n^+m^n—7i^; 
x^  -  3x  4-  2. 

7.  xy  —  2n  +  1;  m  —  n  —  p  —  q;  a^  —  2a^ -\- 2x  —  3; 

1  +  X  -}-  x'^  +  x^;  x+Sy+2a  —  b. 

8.  2a^-Sa^-x  +  3;    x  -  2y  -  Zz  +  2n\  wT  ^-  tT 

9.  ^a-2b-V\c',xf-y-  +  \a-\b;la^-x  +  l', 

10.  l^lx-\x',la^-\x-\;\ar-\a-+\xy', 

2  a;t  +  5  a:i  +  7. 

11.  '^x^-2x^  +  \x^-x-^;  m^"-f  a;i«2/-t*-|^'^-3; 
2i-3i. 

29.   Any  Power  of  a  Binomial.     It  may  be  shown  by 
actual  multiplication  that: 


INVOLUTION.  57 

(a  +  6)3  =  a3  +  3  an  +  3  a?;^  +  68. 

(a  -  6)3  =  a8  -  3  a26  +  3  a  6^  -  JS; 

(a  +  6)*  =  a*  +  4a36  4-  6  a^l^  +  4:ah^ -^  6*; 

(a  -  6)*  =  a*  -  4  a36  +  6  a2  62  -  4  a68  +  6*; 

(,f  +  6)'^  =  a^+  5a*6+  10  a8  62  +  10  a263  4-  5a6*+6«; 

(,i  -6)'^  =  a6-5a*6+  10a362-  10a263+  5a6*-6fi; 
and  80  on. 

In  each  of  the  above  products  we  obsei-ve  tlie  following 
laws: 

I.  The  number  of  terms  is  one  more  than  the  exponent  of 
the  binomial. 

II.  If  both  terms  of  the  bin^omial  are  positive,  all  the  terms 
are  positive, 

III.  If  the  second  term  of  the  binxrmial  is  negative,  the 
odd  tei-ms,  in  the  product,  are  positive,  and  the  even  terms 
negative. 

IV.  TTie  first  and  the  last  terms  of  the  product  are  respec- 
tively the  first  aiul  the  last  terms  of  the  binoinial  raised  to 
the  power  to  which  tlie  binomial  is  to  be  raised. 

V.  The  exponent  of  tlie  first  tei^m  of  the  binomial,  in  the 
second  term  of  the  product,  is  one  less  than  the  exponent  of 
the  binomial,  and  in  each  succeeding  term  it  decreases  by  one. 

The  exponent  of  the  second  term  of  tlie  binomial,  in  the 
second  term,  of  the  product,  is  OTie,  and  in  each  succeeding 
term  it  increases  by  one. 

Thus,  omitting  coefficients, 

(a  +  6)«  =  a«  +  a^b  +  a*62  +  aS^s  ^  «2  j4  j^  ab^  ^  6« 

VI.  The  coefficient  of  the  first  and  the  last  term  is  one, 
that  of  the  second  term  is  the  exponent  of  the  binomial. 


58  ELEMENTS   OF  ALGEBRA. 

The  coefficient  of  any  term,  multiplied  by  the  exponent  of 
the  first  term  of  the  binomial  in  that  term,  and  divided  by 
the  number  of  the  term,  will  be  the  coefficient  of  the  next  term. 

Notes :  1.  The  sum  of  the  exponents  in  any  term  of  the  expansion  is  the 
same,  and  is  equal  to  the  exponent  of  the  binomial . 

2.  The  coeflScients  of  terms  equally  distant  from  the  first  terra  and  the  last 
term  of  the  expansion  are  equal.  Thus,  we  may  write  out  the  coefficients  of 
the  last  half  of  the  expansion  from  the  first  half. 

If  one  or  both  of  the  terms  of  the  binomial  have  more 
than  one  literal  factor,  or  a  coefficient  or  exponent  other 
than  1,  or  if  either  of  them  is  numerical,  enclose  it  in 
parentheses  before  applying  the  principles.     Thus, 

Example  1.    Expand  (2x'^-5a^xy 

Process. 

(2  a:8  -  5  a2 a;)*  =  [ (2  a;3)  -  (5  a2 x)  ]4 

=  {2x^y~4{2x^f{5a^x)-\-6{2xy{5a^x)^-4{2x^){5a^xy 

+  (5a^xy 
=  24a;i2_4  X  2^x^  X  5a^x-\-6X  2^x^  X  b^a^x^-  4  X  2a:8 
X5^a^x^  +  5^a^x^ 

=  16a;i2_4X8a:«X  5a^x-{-6  X  4x«X  25a*a;2-4x  2x8 

X  125a^x^+626a^x^ 
=z  16a;i2_160a2a;io  +  600a4a:8_1000a6x«+625ft8a:* 

Explanation.  In  the  expansion  the  odd  terms  will  be  positive, 
and  the  even  terms  negative.  The  first  term  is  (2  x^y,  and  the  fifth 
or  last  is  (5  a^xy.  The  exponent  of  (2  x^).  is  4,  and  in  each  succeed- 
ing term  it  decreases  by  1.  The  exponent  of  (5  a^x)  is  1,  and  in  each 
succeeding  term  it  increases  by  1.  The  coefficient  of  the  second  term 
is  4.  For  the  second  term  we  take  the  product  of  4,  (2  x^)^,  and 
(5  a^x).  To  find  the  coefficient  of  the  third  term,  we  multiply  the 
coefficient  of  the  second  term  4  by  3  (the  exponent  of  (2  x^)  in  that 
term),  and  divide  the  product  by  2  (the  number  of  the  term),  and 
have  6.  Hence,  the  third  term  is  6  (2  x^y  (5  a^x)''^.  The  coefficient 
of  the  fourth  term  is  found  by  multiplying  6  (the  coefficient  of  the 
third  term)  by  2  (the  exponent  of  (2  x^)  in  the  third  term),  and 


INVOLUTION.  59 

dividing  the  product  by  3  (the  number  of  the  term).  Hence,  the 
fourth  term  is  4  (2  x*)  (5  a*  a:)*.  Performing  operations  indicated,  we 
have  the  required  result. 

Example  2.     Raise  1  —  §  x"  to  the  fifth  power. 

Process. 

(l-§x")»  =  (l)»-5(l)*(|x«)  +  10(l)»(§x»)«-l0(l)«(fz»)»  +  5(l)(fxn)* 

=  1»-5X  l*x|a*+10X  l«x|x2--10X  l«X^x»«  +  5X  1 


Exercise  22. 
Expand  and  simplify  the  following  expressions : 

1.  (a  -  6)7;  {a  4-  x)^;  {a^  ~  ac)*;  (a^  -  4)^;  (2  +  a)*; 
(«-l)^  (1  -aY;  (2a-Sby. 

2.  (xh  -  3)*  ;  (ax-S  x^f ;  {x  -  3)^  (2  a^z  +  3  62^)8; 
(2ax-\-  Sbyy. 

3.  («  +  2)e;  (a-2)«;  (2-Ja)*;  Cja-36)*;  (Ja  +  }6)*; 
(a  +  6)W. 

4.  (ai  -  2  -  a-^y ;  [(2:  +  y)H  {x  -  yff\  (1  +  a  +  a^)^ 
-  (1  -  a  +  2  a2)'f 

5.  (a  +  2i)*-(a-26)*;  (3  -  2a  +  a2)2  -  (2  -  a)*; 
(3i  +  5i)2  _  (2i  -  3i)2 

Queries.  How  prove  (—  m)*  =  ±  m",  according  to  the  value  of 
n,  whether  an  even  or  odd  integer  ?  How  prove  the  method  for 
squaring  any  polynomial?  How  prove  the  laws  for  raising  a  bino- 
mial to  any  power? 


60  ELEMENTS  OF  ALGEBRA. 

CHAPTER  VI. 

ALGEBRAIC  DIVISION. 

30.  Division  is  the  inverse  of  multiplication,  and  is  the 
operation  of  finding  the  other  factor,  when  a  product  and 
one  of  its  factors  are  given.  The  product  is  now  called  the 
Dividend,  the  given  factor  is  the  Divisor,  and  the  required 
factor  is  the  Quotient.     Thus, 

since  a^  X  a^  =  a^  .*.                  a^  ^a^  =  a^-^ 

since  a-^Xa^^a^,  .*.                   a^-^a^  =  a-^\ 

since  a^  X  a-*  =  a,  .-.                  a-^a-*  —  a^; 

since  a"*-"  X  a"  =  a"*,  .*.                  a"» -r- a"  =  a"»-"; 

since  «"»+"  X  a-"  =  a"*,  .*.               a™  -^  a-"  =  »*"+" ; 

since  Sa^ft*  x  2a-26  =  6a6^  .-.  Qah^ ^^a-H  =  ^a^b^; 

since  9a-362  x  3a*65  =  27a6^  ...  ^1  aU' -^Za^h^  =  ^a-H^; 

since       5  a*6"~^  X  4a~*6^  =  20aH*,  .•.  20a^h^^Aa~^h^  =  baH~^ ; 
etc.     Hence,  in  general, 

To  Divide  a  Monomial  by  a  Monomial.  To  the  quotient  of 
the  numerical  coefficients  annex  the  literal  factors,  each  taken 
'with  an  exponent  obtained  hy  subtracting  its  exponent  in  the 
divisor  from  its  exponent  in  the  dividend. 

Illustrations. 

a^h^c^m^-^  a^b^c^m^  ap-^b'^-^c^-'^m^-'^  =abc*m. 
63a-26V5^  7«-36c'»  =  9a-2+3i2-ic5-4     =()abc. 

1^    2A2  .  «^  5a2i2-i  ^a^b  ,.   ^   „ 

I5a^b^-^6bc         = — = (Art.  2). 

2c  2  c    ^  ^ 


ALGEBRAIC    DIVISION.  61 

Exercise  23. 


Divide : 


1.  3a362  by  ab]  I6a*i^  by  3aH^;  20a^}^c^  by  5a6V; 
Smi  by  5  m^. 

2.  tT^  by  ?i~i^;  a*  by  a*"^;  a^j-s^n  y^y  ^35-2^2.  ^m+i. 
by  a"*  —  ;  2'+'  by  2'-'. 

3.  15  a-t  6-i  a^  by  9  a-2  6"!  ic^.  ^  ^i  fti  by  f  (A  ji  ; 
21a*m2ic'  by  Tama:*. 

4.  24a"j9'"  by  3a>";  36a'"mV^  by  9amyri*;  «*'+y-* 
by  a^/. 

5.  (x  -  y)6  by  (a;  -  y^;  (a  -  c)*+8  by  {a  -  c)*-i ; 
|fe*^^•-  by  f  6/H^. 

6.  (6a3  62,;  X  iSftSJV)  by  (S^a^'c^  x  2a*c8);  a*"'  by 
a*";  (2  77i7i")2*  by  (2mn«j^ 

31.  Only  a  positive  number,  +  a,  when  multiplied  by  +  6,  can 
give  the  positive  product  +a6.  Therefore,  +ab  divided  by  +6  gives 
the  quotient  +  a. 

Thus,  since  aX6  =  +  a6,  .•.  4-a6-f  +  6  =  -f-a; 

since      aX-6  =  —  a6,  .*.  — a6^ —  6  =  -|-a;  , 

since      — aX6  =  —  a6,  .*.  — a6-^  +  6  =  —  a; 
since  —  aX— 6  =  4-a6,  .*.  -{-  ab-. —  b  =  —  a. 

Hence,  in  general, 

Law  of  Sig^.  If  the  dividend  and  the  divisor  have  the 
same  siguy  the  quotient  is  positive.  If  they  have  opposite 
signSy  the  quotient  is  negative. 


62  ELEMENTS  OF  ALGEBRA. 

Example.    Divide  12a"»  by  —  4  a". 

Solution.  Since  there  is  a  factor  4  in  the  divisor,  there  must  be 
a  factor  3  in  the  quotient,  in  order  to  give  a  product  of  12  in  the  divi- 
dend. Since  there  are  m  factors  of  a  in  the  dividend,  and  n  in  the 
divisor,  there  must  be  m  —  n  factors  of  a  in  the  quotient,  in  order  to 
give  a  product  of  a"»  in  the  dividend.  Hence,  120'"  -f  -  4  a*^  =  —  3  a"*-", 
because  only  a  negative  number,  — Sa"*"",  when  multiplied  by  —4a" 
can  give  the  positive  product,  12  a*". 

Illustrations. 

-  ISa^mHS-f  3a2m*62  -  -  b  a^-^b^-^m^-'^  =  ~5a^bm^. 

-  5  x^^y^z^  i-  -10  x^y^z^  =  +  ^  cc^o-s  ^^8-5^6-3  ^  ^  ^x^y-^z\ 

ia'{a-by{x  +  y)^^  -4:a{a-b)^{x  +  yy=  -|a"-^(a-6)  (x-t-i/)"*-". 


Exercise  24. 

Divide : 

f.    6^  by  3^;  -20aH^cJ  by  lOahc;  35a^^hy-7a^; 
-laHc^  by  -7aHc^. 

2.  27ax*  by  -9^4.  _|a6^,6c6  by  ^aHc'^;  .^aH^^c^^ 
by  fa^^iici*;  12^2"?/ 2  by  -f^j''?/. 

3.  Z\7n?n^3iP-  by  -2i  m-i7i-3^-2;  -  5|  m-^j^-iyio  by 
*lj2ga2m3x-4?/;  3.2Jrt-s^?/S  by  2.Qt2\a-^xy\ 

4.  .O^a^wiV^*  t)y  -.0|a2^2/2^3.  _9.3m3«^2-^«-32/§c 
by  .3m3«+ix'*-4  2/K 

5.  .^x'^ifhy-ix'y-'^'^-  -J(a     fe)3c8  by  .6(a-&)2ci0; 
~  .3ai'"ft^  by   -.2  a"  6". 


ALGEBRAIC   DIVISION.  63 

6.  -  .375  xi  ?/U-^^  -  y^^     by    -  i^  oc^  y  (xi  -  i/l)l . 
8m-Si-^r-02y7  by  9  m    ^  ,.   "- x-^  y- \ 

7.  -1.2aiO(jc-y)"r»  hy  .^a^{x-yfz^'^\  m-^n^x-yY 
(y-zY  by  w   2«n2''(a;- 7/)  '•(?/- 2)i^. 

Simplify  the  following,   that  is,  perform  the  indicated 
operations : 

X  -~.5a2''62«c-2'. 

9.  {a-H*-^2ab)  x  -2a2fe-2  x  (- .Gaifti -=- -  .3aUi). 

10.  (.3a-'"6-'*c-''H- .03a"'6-c'') -r  1  j  «-3'"6-3»c-3''A:. 

11.  (4§«-»6(ijc-2-f-lia-U-3rf*) 

X  [6  a2c-  irf3  ^  ^  (84  ^8  j8  c  -f-  7  a*  b^  c^)]. 

12.  (ic""^'*  X  a-U-i.i-"-")  x  (rtUa^^-^^^i  -!-6ir*'"7/-t). 

13.  (-Ua^H^c-^^-7aH*c-^) 

-=-  (28  a-^iV  ^  -  4  a-'^b-^c-"). 

14.  (1.7«-i6-^cijc2-M.l«-2i-ia:8)x  (a"'i*c-6^al62c3). 


32.    Since  (a  •^-  b)  m  =  a  m  +  b m,  .-.  (am  +  bm)  -r  m  =  a  +  b. 
Since  (a  -  b)  m  =■  am  —  bm,  . • .  (a m  -  6  w)  f  m  =  a  -  ft. 

Since       (iy-2y«2-3x»ir')  X -3xy»=  -3x«y<  + 6xi/*2  +  9x*^, 
.-.  (-3x»y*4-6xy»2  +  9x<y)T  -3xi/»  =  2r/-2ya2-3x»y-». 
Hence,  in  general, 

To  Divide  a  Polynomial  by  a  Monomial.    Divide  each  term 
of  the  dividend  by  the  divisor,  ami  add  the  results. 


64  ELEMENTS  OF  ALGEBRA. 

Exercise  25. 

Divide : 

1.  2a?+  ^a^y  ~  8a^y^  by  2a^-  21  m^n^  -  1  m^n^ 

—  14  myi  +  63  by  7  mn. 

2.  a^hc  --  a?  b^  c2  -  a^  W'  c^  +  a^  b^  c^  by  a^bc;  42  a^ 

—  1.1  a;2  +  28  ^  by  .7  x, 

3.  28a3  +  9a2-21a  +  35  by  7a;   4:  a^  b^  -  16  aH^ 
+  4:aH^  by  -  4  aH^. 

4.  66t262c3  _  48^264^2  +  36^2^2^!  _  20abc^  by  4a&ca 

5.  2.4  m27i2  —  .8  mht^  —  2.4  m  ?i2  +  4:m^n^  by  .8  m  n ; 
icf  —  x^  y^  by  ici. 

6.  -"^a^^^ab-^ac  by  -1.5a;  .5  m57i2_  3^3^*  ^y 

—  1.5  m^v?. 

7.  -  72  o^  c2  -  48  a}  ^10  +  32  ^2  c^   by   16  a^  c^;    3.6  n* 

—  4.8  rtf  by  4  n\ . 

8.  11^2^3^  3  a;  71 -2.4  2/2  by  .'^xy\  .09  7^1*- 2.4 m^ 71 
+  4.8  «i5  by  .03  wi* 

9.  -  ft"*  +  2a-"'  -  3a"  by  -  a^\  m"+i-  7/i"+2  +  77i'*+3 

—  m"+*  by  m?. 

10.  2.1  a  2:2  y"*  4-  I.4a3;:c4^"- 2.8a5^2^P  by  -  ,1  axy"". 

11.  a'"53_^-+ij2_|_^n-2j  by  -ab;  -2a^a^-j-S.5a^x* 
by  2.3^  a^x, 


ALGEBRAIC   DIVISION.  65 

12.  2.25  a^x  -  .0625  aire  -  .375  a  ex  by    .'67^  ax; 
llxi-33a;*  by  11  x*. 

13.  72  wt  -  60  mi  ni  +  12mi  ni  -  6mT^ni  by  24  mi. 

14.  36  (x  -  yf  -  27  (x-y)3  +  lS{x-y)  by  9(a:-y). 

15.  -123ry*2'S0sf''-^^y^z+10SaP^y2f-^^  by  -6^y*^ 

16.  m"  (x  —  y)«  —  7;ia(a;  —  t/)"  by  m*  (a;  —  y)*. 

17.  {x-hyY(x-yy-\-(x+yy{x-yy  by  (x-^yy{x-y)\ 

18.  -2.5m2+1.6m7i+3.3m  by  -.83m;  a-^i-ai^ift^+a ^ 
by  a-w. 

33.    It  may  be  shown  by  actual  multiplication  that : 

(TO+n+/>)  (x+y-f-z)  =  ma:+TOy+m2  +  nx  +  nT/  +  nz+;)a;+/>y+/)2. 
.♦.  (nu:^-my+'nu-\-nx-\rny-]rnz-\-px+py-\-pz)^{x  +  ]i  +  z)  =  m-\-n-\-p. 

The  division  is  performed  as  follows : 

Separate  the  dividend  into  the  three  parts  mx  +  my-fmz, 
n  X  4-  n  y  +  n  2,  and  px  +  py-\-pz.  The  first  term  of  the  (quotient, 
m,  is  found  by  dividing  m  x,  the  first  term  of  the  dividend,  by  x,  the 
first  term  of  the  <U visor  ;  multiplying  the  entire  divisor  by  m  will 
produce  the  Jirst  part  of  the  dividend.  The  second  term  n  of  the 
quotient  is  found  by  dividing  the  first  term  of  the  second  part  of  the 
dividend  by  the  first  term  of  the  divisor  ;  multiplying  the  entire  di- 
visor by  n  will  produce  the  second  part  of  the  dividend.  The  third 
term  p  of  the  quotient  is  found  by  dividing  the  first  term  of  the  third 
part  of  the  dividend  by  the  first  term  h{  the  divisor ;  multiplying  the 
entire  divisor  by  p  will  produce  the  third  part  of  the  dividend.  The 
work  is  conveniently  arranged  as  follows ; 

5 


(    ^MV£RS(Ty\ 


66 


ELEMENTS  OF  ALGEBRA. 


g 

+ 

g 

■v..^ 

w 

M 

f    M 

N 

^i 

fts 

a, 

^1 

4- 

+ 

4- 

+ 

?=^ 

5>i 

^ 

5ss 

a^ 

a, 

a, 

a^ 

+ 

+ 

+ 

+ 

H 

« 

H 

H 

a. 

^H 

^i 

a, 

4- 

+ 

M 

l^ 

?\i 

$ 

8 

s 

+ 

+ 

+ 

?s» 

5rj 

?s^ 

S 

S 

S 

+ 

+ 

+ 

« 

H 

H 

s 

s 

S 

4- 

N        N 

fc    s 

+  + 

>i    a>i 

S     g 

+    + 

«     « 

1.  ^ 

N 

^*" 

+    ^- 

>.  -g 

•+3 

d 

■^ 

d 

quotie 

O 

'S 

-l 

a 

^^"^ 

"o 

1 

f 

1 

1 

-•^ 

^ 

o 

•S 

^, 

^3 

o 

^ 

05 

'^ 

^ 

^ 

^ 

'^u 

c 

^ 

si 

i=l 

d 

a 

^ 

rt 

13 

3 

f-i 

rt 

;-• 

F-i 

o 

o 

d 

O 

w 

Tl 

•iH 

'eS 

^ 

% 

_> 

'> 

'S 

§ 

'S 

OJ 

^ 

O) 

(B 

2h 

O) 

^^ 

_t- 

J 

c 

"3 

rt 

d 

O) 

cj 

<u 

(u 

%-i 

■^ 

V-( 

n:J 

^_i 

o 

B 

o 

d 

c3 

o 

-u 

S 

-M 

-u    * 

o 

o 

"73 

o 

:=i 

|3 

rt 

d 

-ij 

g 

^ 

n:) 

o 

o 

Q 

o 

&4 

;h 

PU 

S 

fl^ 

m 

PU 

+ 

+ 
CO 

I 
I 


^         I 

I  7 

I 

CO 

I 


+  + 

CM 


H 

H 

1 

CO 
1 

1 

1 

1— 1 

4- 

H 

(M 

4- 

1i 

CO 

1 

1 
1* 

1 

1 

1 

1 

« 

1 

1 

CO 
1 

1 

^ 

1 

1 

1 

1 

1 

(?q 

-t 

CO 

CO 

+ 

1 

4- 

4- 

"« 

TO 

CO 

CO 

+ 

+ 

Tj 

^ 

t, 

(M 

(M 

1 

i 
1. 

1. 

1 

1 

.  CM 

1 

1 

fj" 

o 

.2 
a> 

f 

^ 

J 

1 

m" 

*> 
^ 

d 

0) 

.^ 

d 

•5 

1 

"-I3 
d 

0) 

d 

■^3 

% 

^ 

^ 

+3 

S 

S 

i 
1 

(N 

CO 

•1-1 

1^ 

ptH 

1 

OQ 

1 

ALGEBRAIC  DIVISION.  67 

Explanation  Dividing  the  first  term  of  the  dividend  by  the  first 
term  of  the  divisor,  we  have  a:*,  the  first  term  of  the  quotient.  Now 
as  we  are  to  find  how  many  times  x*  —  3a:^-j-2z+lis  contained  in 
the  dividend,  and  have  found  that  it  is  contained  a:*  times,  we  may 
take  X*  time.s  the  divisor  out  of  the  dividend,  and  then  proceed  to  find 
how  many  times  the  divisor  is  contained  in  the  i-emainder  of  the  divi- 
dend. Dividing  the  first  term  of  the  remainder  by  the  first  term  of 
the  divisor,  we  have  —  2  a:,  the  second  term  of  the  quotient.  Simi- 
larly, we  find  the  third  term  of  the  quotient.  Hence,  the  quotient  is 
x«  -  2  X  -  2. 

Notes:  1.  Algebraic  division  is  strictly  analogous  to  *Mong  division*'  in 
Arithmetic.  The  arrangement  of  the  terms  corresponding  to  the  order  of  suc- 
cession of  the  thousands,  hundreds,  tens,  units,  etc.,  and  the  processes  for  both 
are  exactly  the  same. 

2.  It  is  convenient  to  arrange  both  dividend  and  divisor  according  to  poioers 
of  the  same  letter  ascending  or  descending. 

3.  It  may  happen  the  division  cannot  he  exactly  performed  ;  we  then  alge- 
biaicaUy  add  to  the  quotient  the  fraction  whose  numerator  is  the  remainder, 
and  whose  denominator  is  the  divisor.  Thus,  if  we  divide  x"^  —  2xy  —i^  by 
X  —  y,  we  shall  obtain  x  —  y  in  the  quotient,  and  there  vnll  be  a  remainder 

—  2ya.     Hence,  (xS  -  2xy  -  y^)  -r  (x  -  y)  =  x  -  y  -  ~~.  ■ 

X      y 

Example  2.     Divide  a*  +  6'  -I-  c»  -  3  a  6  c  by  a  +  6  -f-  c. 
Arranging  acconling  to  the  descending  powers  of  a,  we  have: 
Process.  Dirisor.  Diridend.  Quotient 

a-f6+c)a»  -3a6c-|-6»H-c»(a*-<26-ac 

a*  times  the  divisor,  a»-t-a^fe-fa^ 

First  remainder,  -aV)-a*c  -3a6c+6»4-c» 

—  ab  times  the  divisor,         —aV*        —ab^ —  abc 

Second  remainder,  -a^(H-ab^        -2a^c+6»+c» 

—  ac  times  the  divisor,  — q^c         —ac^—  ahc 

Third  remainder,  ah^^ac^  abc-{-l^-\-c* 

b^  times  the  divisor,  ab^  ■       -\-b^-\-b^c 

Fourth  remainder,  ac*-  abc-b^c+c* 

c*  times  the  divi.sor,  ac'  -^bc^-c* 

Fifth  and  last  remainder,  -abc-b^c-bc^ 

—  be  times  the  divisor,  —abc—b^c-bc^ 

To  verify  the  work,  multiply  the  quotient  by  the  divisor. 


68  ELEMENTS  OF  ALGEBRA. 

Example  3.     Divide  i^  xy^  +  \x^  +  ^y^  by  ^y  +  ^x. 

Process. 

^x  +  ^y)\x^  +^xy^+^y^{j^x^-^xy  +  {y 

Divisor  X  ^x\  ^x^  +  ^x^y 

First  remainder,  —  i  ^^2/  +  -^  ^  2/^  +  i^y^ 

Divisor  X  —  ^xy,  —  ^x'^y  —  ^  x y^ 

Second  and  last  remainder,  i  ^  ^^  +  iV  2/* 

Divisor  X  \  y%  i  ^  J/"  +  i^  y^- 

Hence,  in  general, 

To  Divide  a  Polynomial  by  a  Poljmomial.  Divide  tJie  first 
term  of  the  dividend  hy  the  first  term  of  the  divisor  for  the 
first  term  of  the  quotient;  multiply  the  entire  divisor  hy  this 
term,  and  subtract  the  product  from  the  dividend.  Divide 
as  before,  and  repeat  the  process  until  the  work  is  completed. 

Exercise  26. 

Divide : 

1.  14  x^  +  45  ^2/  +  "^8  x^i/  +  ^bxf  +14.y^  hy  2x^ 

+  5  xy  +  7  y\ 

2.  x'^  —  2x^y+2x^y^  —  xy^  by  x  —  y;  a^  —  2ah^-^h^ 
hj  a-b.  ^ 

3.  f-5y^  +  9  2/-6y^-y+2  by  y^  -  3  y  +  2  ; 
y^-1  hy  y-1. 

4.  x"^  +  xy  +  2  xz—2y^+ 7  yz  — 3  z^  hy  x  —  y +3z; 
c^  —b^  by  a  —  h. 

5.  2^2/+  36?/  +  10fe;:c  +  15  62  by  2/  +  5&;  a6  +  a^b 
hy  a  -\-b. 

6.  .125ic3-2.25a;22^4.  \Z,^  xy^ -27 y'^  by  .^x-3y, 

7.  ?/-62/^-2^  +  54^-3a;2?/  by  2x-y\  x^-y^  by  x  +  y. 


ALGEBRAIC  DIVISION.  69 

8.  a^t/^—a^  —  y^+lhyxy  +  x-\-i/+l;4:i/  +  4:y 
-y8  by  3y  +  2i/2  4-2. 

9.  s^+i^  —  z^-{-3xyzhyx  +  y  —  z;  x—y  by  x^  —  yi. 

10.  3^y^-\-2xi/^Z'-a^z^-{-i/^z^   by   xy  +  xz  +  yz; 
x^  —  y^  hy  xi  —  yi. 

11.  12x*-26a^y-Sx^f+10xf-Sy^hySx^ 
-2xy-\-f. 

12.  a^^'f^-Zxy-\hyx-\-y-\)    5V  ^  "  iV  ^ 
+  iV  ^  -  S*I  by  J  X-  -  i. 

13.  12  ;c«  ?/9  -  14  x*  2/^  +  6  a;2  ^9  _  ^9  by  2x^y^-f) 
a^  —  y^  by  a;«  —  y^. 

14.  a^  6  -  a  ^  by  a3  +  63  +  a  ^2  +  ^2  J  .  ^  ^  ^4  _  -^ 
—  x~^  by  a;  —  x~^. 

15.  a^  +  :r*  y  +  a:^  2/*  +  i^  ?/^  4-  a:  ?/*  +  7^5   by    a;^  +  7/^ ; 
al  —  6f  by  a^  —  fci 

16.  Ja3  + Y«^-l-25a  +  2.25  by  Ja  +  3;  .Ibx^y^ 
+  .048  a:^  by  .2  a;^  ^   5^,^ 

17.  at-a2-4at  +  Ga-2ai  by  J-4«i  +  2;  a:S_y5 
by  a;  — ;/. 

18.  a;8  +  7^  +  23  4.  3  ^3y  ^.  3  aj^J  by  a;  +  y  +  2 ;  .5  a;8 
+  a.^+  .375  a:  +  75  by  J  a;  +  1. 

19.  x^-\-^y^^-^-^xyzhy  3^-\-^f^7?-xz-2xy 
-2yz. 

20.  ^g  a:*  -  I  a,-3  -  J  r^  +  I  a:  +  Jg^  by  1.5  a:^  _  3,  _  | 

21.  ofi  —  1^  hy  a^  +  Qc^  y  -\-  X  y^  ■\-  ij^ ,    a^  —  7^    by    a^ 
+  a;  y  +  2/2. 


70  ELEMENTS  OF  ALGEBRA. 

22.  10 a^--27aH  +  34:a^I^-lSah^-8b^hj  5 a^-6ab 
-  2R 

23.  36x^+^i/+.25-4:X7j-6x  +  ^y  by  Qx-^-.d. 
24    ai2  +  2  aH^  +  Z^^^  by  «*  +  2  ^2  ^2  +  ^,4.  ^6_2,6  ^y 

a^-2a^b  +  2ah^-h\ 

25.  2  ^^"  —  6  iz2«  y-  +  6  rr"2/2"  -  2  ?/3«  ^y  2^  —  2/«;  ^» 
+  2/3"  by  it"  +  ?/. 

26.  i6'2«— 7/2'"  +  2  7/"*^'- s2'  by  ^''+ ?/"•  — ;3';  32^-22^ 
by  3^  -  2". 

27.  ifX^-lil-xi/^  by  f^-.752/;  a:-t"'-3aj-i"'2/-i'' 
+  2  y-h  by  a-  ^"^  —  2/-i". 

28.  ?/2a;2m^2  7/22:'"+"+  2  2/r  a?"*  +  22a;2n  +  2ra?";2  +  r2 
by  3/  a:*"  +  2;  ^"  +  7\ 

29.  a;~i  +  2x~^y~i  +  2/"^  by  a?~^  +  y~^;  x^  +  ?/*  by 
a?2  +  22  a?  2/  +  2/2, 

30.  x~'^  —  y'~'^-\-2y~^z~^  —  z~'^  by  a;~i  +  2/~^~^~^j 
a?*  —  3  2/*  by  x  —  y. 

34.  There  are  special  methods  for  finding  the  quotient 
of  binomials,  hy  inspection,  which  are  of  importance  on  ac- 
count of  their  frequent  occurrence  in  algebraic  operations. 
Thus, 

It  may  be  shown  by  actual  division  that : 

a—b  ^a—b 

^^—^  =  a^+as  6+a2  h^+ab^+b^ ;  ^"—^  =  a^  +  a'^b  +  a%'^ + 02^8 +ab^+b^) 
a—b  a—b 

and  so  on.     Hence,  in  general,  it  will  be  found  that, 


ALGEBRAIC  DIVISION.  71 

The  difference  of  any  two  equal  powers  of  two  numbers  is 
divisible  by  the  difference  of  the  numbers. 

In  each  of  the  above  quotients  we  observe  the  following 
laws : 

I.  The  number  of  terms  is  equal  to  the  exponent  of  the 
powers. 

II.  Hie  signs  are  all  positive. 

III.  The  exponent  of  a  in  the  first  term  is  one  less  than 
the  exponent  of  a  in  the  first  term  of  the  dividend,  and  in 
each  succeeding  term  it  decreases  by  one  {in  the  last  term  its 
exponent  is  0,  or  a.  disappears). 

The  exponent  of  b  in  the  second  term  is  one,  and  in  each 
succeeding  term  it  increases  by  one  {in  the  last  term  its  expo- 
nent is  one  less  than  the  exponent  of  b  in  the  dividend). 

IV.  The  first  term  is  found  by  dividing  the  first  ter^n  of 
the  dividend  by  tJie  first  term  of  the  divisor. 

V.  To  find  each  succeeding  term,  divide  the  preceding 
term  by  the  first  term  of  the  divisor,  and  multiply  the  restUt 
by  the  second  term  of  the  divisor  regardless  of  sign. 

Example.     Divide  1  —  w*  by  1  —  n. 

Solution.  Dividing  1,  the  first  term  of  the  dividend,  by  1,  the 
first  term  of  the  divisor,  we  get  1  for  the  first  term  of  the  quotient 
Now  divide  the  first  term  of  the  quotient  by  the  first  term  of  the 
divisor,  and  multiply  the  result  by  n,  the  second  term  of  the  divisor 
(regardless  of  sign),  for  the  second  term,  n,  of  the  quotient.  Dividing 
the  second  term  of  the  quotient  by  the  first  term  of  the  divisor,  and 
multiplying  the  result  by  n,  we  have  n^  for  the  third  term  of  the  quo- 
tient. Similarly,  we  find  n*,  and  n*  for  the  fourth  and  ffih  terms, 
respectively.     .*.  (1  —  n*)  -r  (1  —  n3  =  1  -|-  n  -f  n*  +  n*  +  n*. 


72  ELEMENTS  OF  ALGEBRA. 

Exercise  27. 

Divide  by  inspection : 

1.  m^  —  n^  by  m  —  n\  a^ m^  —  h^n^  by  am  —  hn\ 
m^n^  —1  by  mn  —  1. 

2.  l—m^n^a^hjl—mnx-jix yj*—  {x zj  hy  xy  —  xz; 
1  —  a''b''  x'^  hj  1  —  ahx. 

In  order  to  apply  this  principle  the  terms  of  the  divi- 
dend must  be  the  same  powers  of  the  respective  terms  of 
the  divisor.  It  is  not  necessary  that  the  exponents  of  the 
terms  of  the  divisor  be  1,  nor  that  they  be  the  same,  nor 
that  the  exponents  of  the  terms  of  the  dividend  be  the 
same.     Thus, 

Example      Divide  x^^  —  y^^  by  x^  —  y*. 

iSolution  Dividing  x^^  by  x^,  we  have  x^  for  the  ^rst  term  in  the 
quotient.  Now  divide  x^  hj  x^  and  multiply  the  result  by  y*,  for  the 
second  term,  x^  y*,  in  the  quotient.  In  like  manner  we  find  x^y%  and 
y^^  for  the  third  and  fourth  teims  of  the  quotient. 

. •.   (a;i2  -  i/16)  -i-  (x3  -  2/4)  =  a;9  +  ^6  ?y4  +  a:3  2/8  +  ^12 

So  in  general  x"*  —  y"^  divides  2:""  —  ;v""*  {n  being  any 
positive  integer),  since  the  dividend  is  the  difference  be- 
tween the  nVa  powers  of  the  terms  of  the  divisor. 

3.  a^  -  W  by  a^-l^-  x^^  -  f^  by  x^  -  y'^  \  x^^  -  y^'^ 
by  a::^  —  y'^. 

4    ai5  -  &30  ]3y  a3  -  &6  .  x^m  _  ^35n  -^^  ^n  _  ^n .  2io« 

-  a^""  by  22"  -  x"^. 

We  may  easily  apply  these  principles  to  examples  con- 
taining coefficients  as  well  as  exponents;  also  to  those 
involving  fractional  or  negative  exponents.     Thus, 


ALGEBRAIC   DIVISION.  73 

Example.     Divide  81  a^^  -  16  A**  by  3  a«  -  2  h\ 

Solution.  Dividing  81  a"  by  3  a«,  we  have  27  a»  for  the  frst 
term  of  the  quotient.  Now  divide  27  a*  by  3  a*  and  multiply  the 
result  by  2  6*,  for  the  second  term,  18  a®  6*,  in  the  quotient.  Simi- 
larly, we  find  12  a'  6^*,  and  8  h^^  for  the  <At>d  and  fourth  terms  in  the 
quotient. 
.-.  (81a"-16634)^(3a»-26«)  =  27a»+18a«6«+12a«6»2+86". 

If  a  and  h  are  coefficients,  a^a^**  ~  6"^*""  is  divisible  by 

ax'  —  hif^f  since  the  dividend  is  the  difference  between 

^1        -1 
the  Tith  powers  of  ax*  and  ////'".     In  general,  a?  "•— y   *" 

na  n$ 

divides  x    '^  —  y   ""  (n  being  any  positive  integer),  since 

a 

the  latter  is  the  difference  between  the  ?ith  powers  of  a;    •* 
and  y   '. 

5.  64ai2_27?i»  by  4a^-Sn^;  16  2^  y^^"*  -  ^  m^z^* 
by  4a:8y6*_  j^^4  2io» 

6.  «8  a^S"  -  68  2/3«  by  a2aJJi'  _  ^^  ^^^ .    32  ar^o  -  243  y^ 
by  2  0^2  -  3  3/3. 

7.  x~^  —  y~i    by   a;~i  —  y~i  ;    3^  —  y^    by   a;^  —  yi  ; 
a  xi  —  6i  y  by  ai  x^  —  fci  yi 

35.    It  may  be  shown  by  actual  division  that : 

ai—l)i  a*-h* 

— r-j-  =  a  —  b;  — —r-  =  a*—a^b-{-ab^—b*;  and  so  on. 
a+o  a  +  o 

Hence,  in  general,  it  will  be  found  that, 

The  difference  of  any  two  equal  even  powers  of  two  num- 
bers is  divisible  by  the  sum  of  the  numbers. 

In  each  of  the  above  quotients  we  observe  the  laws  are 
the  same  as  in  I.  and  III.,  Art.  34  ;  also, 


74  ELEMENTS  OP  ALGEBRA. 

VI.    The  signs  are  alternately  +  and  — . 

Hence,  the  principle  may  be  applied  to  different  classes 
of  examples  as  in  Art.  34.     Thus,  in  general, 

If  a  and  l  are  coefficients,  ax^  -{■  h'lf  divides  a"  x"*^ 
—  If  y^"^  (n  being  any  even  and  positive  integer ;  also  m  and 
p  may  be  integral,  fractional,  or  negative),  since  the  divi- 
dend is  the  difference  between  the  nth.  powers  of  a  x^  and 
by-. 

Note,  'rtie  difference  of  the  squai'es  of  two  numbers  is  always  divisible  by 
the  sum  and  also  by  the  difference  of  the  numbers.  Thus,  06  —  68  jg  divisible 
by  aS  ±  64.  jn  general,  a^n  —  IZm  [§  divisible  by  a«  ±  **"  when  n  and  m  are 
integral.    This  is  the  converse  of  Art.  26. 

Exercise  28. 

Divide  by  inspection : 

1.  625  a^x^  —  81  ra^n^  by  5ax+  3  mn;  a^  —  h^  hy 
x:^  +  b^;  x^  —  1  by  x  +  1. 

2.  x^  -  yi  by  x^  +  y^ ;  256  ir*  -  10000  by  4£c  +  10^, 
3    ^im^yin  by  a;"*  +  2/" ;  tV  ^^  -  -^^^^  2^^  by  J  rr^  +  .22/1 ; 

^10    _    ^,10     ]^,y      a,     +     6. 

4.  729  ai2  _  64  ^,18  ^y  S  a^  +  2  b^ ;  a^  x^""  -  ¥y^"*  by 

a  ^-^  +   &2  y2  m 

5.  a""^  —  x'"^  by  a~^  +  a;"^;  a^  x~^  —  b^y~^  by 
a  a:~3  +  h's  y~^-  • 

6.  xh  -  yi""  by  2;i*^  +  t/t^"*;  81a-i"a;  -  Jq  &f«^-f'» 
by  3  a-T5'*a;4  +  |  b^y'^'^. 


ALGEBRAIC  DIVISION.  75 

36.    It  may  be  shown  by  actual  division  that : 

j-  =  a^—ab  +  b^',  j-  =  a*—a*b+a^b^—ab^-^b*;  and  so  on. 

a-\-b  a  +  b 

Hence,  in  general,  it  will  be  found  that, 

The  sum  of  any  two  equal  odd  powers  of  two  numbers  is 
divisible  by  the  sum  of  the  numbers. 

In  each  of  the  above  quotients  we  observe  that  the  laws 
are  the  same  as  in  Art.  35. 

Hence,  the  principle  may  be  applied  to  all  the  different 
classes  of  examples  as  in  Art.  34.     Thus,  in  general, 

If  a  and  b  are  coefficients,  aaf-^-hy"^  divides  a"  a:"' 
_j_  i^ynm  ^^^  beiug  ani/  odd  and  positive  integer,  also  m  and 
p  are  integral,  fractional,  or  negative),  since  the  dividend 
is  the  sum  of  the  nth  powers  of  aaf  and  by^. 

Exercise  29. 

Divide  by  insj^ection : 

1.  x"  ■\-  If  by  X  -\-  y  \  x~^  +  y~^  by  a;"^  +  3rM 
1024  a:«  +  243  y«  by  4  a;  +  3  y. 

2.  128  x^^  +  2187  y"  by  2  2:3  +  3  y2 .  243  x^^  +  32  ^o 
by  3  a:8  4.  2  y2. 

3.  a;!*"  +  2/21*"  by  3^^  ■¥  y^^  -  A:2i«  4- 7^86-  ^y  l^^^m^''', 
a"  +  6"  by  a  +  6. 

4.  wi  w-  +  a;  y  by  wi  ?ii  +  xi  yi  ;  x^  +  ?/V  by  x^  -\~  yi\ 
^\-\-  y'^  by  a;-i  +  y-K 

5.  ax\^b^yhya\x\^biiy\',  (# -f  ( J)  by  (f )i  +  (f )i ; 
a^  +  6^0  by  a  +  62. 

Note.  Since  a«  and  Ifi  are  odd  powers  of  a^  and  h^,  n^  +  6«  is  divisible  by 
cfl  +  62.  aW  and  ft"  are  the  5th  powers  of  a^  and  62,  a  10  +  6W  is  divisible  by 
a*  +  6^.  Also,  a»  and  6*  are  the  third  powers  of  a«  and  6«.  Therefore,  a«  -f  6» 
is  divisible  by  a*  +  b^. 


76  ELEMENTS  OF  ALGEBRA. 

6.  a^  +  612  by  a*  +  &^  2:6  +  1  ]3y  x^ +,1 ',  x^^  +  1  by 

a;*  +  1 ;  a27  +  ^27  by  a^  +  h^. 

7.  a;io  +  2/10  by  2;2  +  2/2 ;  a;!^  +  2/^^  by  a^  +  y^ ;  64:  +  x^ 
by  22  4-  2;2 

8.  64  x^  +  729  f  by  22  a;2  +  9  ^^2 .  ^lo  +  __l^  by  x^  +  ( J)2; 

^24  4.  524    by    «§  +  &8  ■ 

9.  ai8  +  Z,i8  by  a^  +  b^  and  by  a^+P;  j^^  x^  +  J^  f 
by  i^^  +  i2/^- 

10.  ^36  +  J36  by  ai2  +  ^,12  and  by  a^  +  J*;  729  a;^  +  1 
by  9  a:2  +  1. 

11.  a;*2  +  2/42  by  x^  +  ^6  and  by  x^^  +  2/^*.  Query.  Is 
it  divisible  by  2;2  +  /  ?  Why  ?  a^^  +  &18  by  a^  +  h^ ', 
^27  +  J21  by  a9  +  &'  ;  a^^  +  515"'  by  a^  +  J^. 

12.  rr^  +  2/^*  by  o^^^  +  ?/i^  and  by  x^  +  /.  Query.  Is 
it  divisible  by  2^2  +  2/^  ?  Why  ?  a^  +  &12  by  a^  _^  &* ; 
aH9  +  m27n36  by  a^h^  +  77i9  7ii2;  ^15^25  _^  m^o^iio  by 
a^  6^  +  m^  7i2 

Find  an  exact  divisor  and  the  quotient  for  each  of  the 
following,  by  inspection  : 

13.  8  +  a^;  ^6  -  6^  8  -  ^;  a:*  -  81;  a^^  -  h^^; 
81  ai2  -  16  h^;  a"^-  625  ;  a^  -  h^ 

14.  aj20  -  2/15;  ^,^5  +  ^5.  ^12  _  ^12  .  ^6  _  1 .  ^-12  _  2,-12. 
a^a^^-h^fp-  a^-'62)  16  «*  -  81. 

15.  Z2w>  -  h^;  81  «8  -  16  b^;  1  -  y'^ ;  a^ x^  +  1000; 
a*  a;*  —  1 ;  a^  +  m^a:^  ;  2^2  7/2  —  81  (x2 

16.  32  ^10  -  243  2>i5.  cIo.tIO'*  -  a^^x^^;  a^"  +  ftQ''^ 
a^x^  +  b^y^"";  c^  x^^  +  6' 2/^". 


ALGEBRAIC  DIVISION.  77 

17.  aj-^"  +  y-e*;  8a^y3  4.  729;  fti2  yi2j»  _  Ji:^yi2». 
cS^^Sp  _  j8^«^  2:4  _  1296;  ^^S"  -  ftio* 

18.  128.^21  +  2187^^*;  256a;i2-81 //«;  a6»65*  +  a:S'y^'; 
1  +  128  a:!*;  «-«'»- d^^-.  ^j-ii^-i-l. 

19.  x^-y-i_aiy-i;  a-i^-S+l;  i^a^x^'^^^^h-'^f^ 
g^  x-t"-. 00032  y-^". 

20.  ii  c"  +  .002432;2/-i;    256  aa;-t~  -  .0081  J-f*; 

Queries.  How  divide  a  monomial  by  a  monomial  ?  Prove  it. 
How  prove  the  method  lor  dividing  a  polynomial  by  a  polynomial  \ 
In  Art.  35,  the  sign  of  the  last  term  ot"  the  quotient  is  — ,  while  in 
Art.  36,  the  sign  ot*  the  last  term  of  the  quotient  is  4-.  Why  is  this  ? 
What  is  the  product  of  a*  and  a-  *  ?  Prove  it.  What  of  a*  and  or*  I 
Prove  it. 

Miscellaneous  Exercise  30. 
Divide : 

1.  a3 6-3+ T^a2 6-2  +  ^1  by  Jafe-a  +  JJ-i;  ari  +  y-i 
by  x~^  +  ir^- 

2.  a-6+  5a-H-i+10ft-36-2-f  i0a-2  6-3  +  bcrH'^ 
+  6-6  by  a-i  +  b-\ 

3.  2  a2  -  a^  -  2  rt  +  1  by  1  -  a^  ;  x  -  ij  hy  x\  -  iji. 

4.  (a  —  6  -  c)"  -  (a  -  6  -  c)"- "•  -  (rt  —  6  —  c)""  by 
(a-  h-cyr 

5.  2  2^3  +  2  y3  +  2z^-  6zyzhy  (x  -  yf  -\-  (y  -  zf 
+  (z-2:)2;  (2^-ff  by  {x' +  xy  +  ff. 

6.  {x^-2yzf-^f7?  by  ^^2-4^2;  (a:+2y)8  +  (y-3«)3 
by  ar  +  3  (y  -  z). 


78  ELEMENTS   OF  ALGEBRA. 

7.   x^  —  x^i/  -\-  xy^  —  2  xi  y^  +  y^  by  xi  —  xy^  -\-  x^y 
-yl 

8.      2  :r3«  _  6  ,jc2nyn  _j_    e  ^n^2n  _  2  fn    ^^    ^n  _  ^n^ 

by  :r"'  -  3  a;'""^ /  -  6  a:'"-^  ?/2«. 

10.  a3"  -  3  a2«  5«  +  3  a«  &2«  _  53^  by  ^«  _  j« .  3  ^-o: 

-  8  a^  +  5  ^3^  -  3  a-3^  by  5  a^  -  3  ^-^ 

11.  6  «'«  +  '  -  23  a^"  +  ^  +  18  a^  -  a^"-^  -  3  a'"~' 
4.  4a3«-3  -  a""-'  by  2  a^''^^  -  5  a^"  -  2a2'*-^  +  a''-'. 

12.  4  at  -  8  ai  -  5  +  10  ^-^  +  3  a"!  by  2  ai2  -  aiV 

-  3  h~l 

13.  6:r^+3-5:2r"  +  '-6  2r^  +  ^+19af-21r^-^  +4,r^-2 
by  2  x^  +  x^  —  4:X. 

14.  6m*-"-*-2  +    m*-"+^    -   22  m^""   +    19  7?t*-"-^ 

15.  6aj^  +  "+'+2f^-"-^^-9af  +  "+lla;^  +  "-'-6;zf  +  "-2 
^^  +  n-3  by  2  2^'^-^'+ 3a;"  +  ^-a;". 

16.  a""*  —  ct"  fe(''-i)'»  —  a<"*-i>"  &"*  +  5"*"  by  a"  -  &**. 

Find  an  exact  divisor  and  the  quotient  of  the  following, 
by  inspection : 

17.  8a^+l;   16  -  81  a^  ;    64  a^  -  8  &3 .     ^34.  iqOO  ; 
^6  -  64 ;  m^  -71^;   1  -  8  ?/3 ;   a^  6^  -  1. 

19.    8  ;x^6  -  27  ?/-^     64  ai2  _  27  ^-9;     243  a^  +  32; 

cSjc^'*  -  ftS^Sm  .    1^  ^4n  _    0016  hh'^',    29"  al2»  +  36n 


EVOLUTION.  79 


CHAPTER   VIL 
EVOLUTION. 

37.  Evolution  is  the  operation  of  finding  one  of  the 
equal  factors  of  a  number  or  expression.  Evolution  is  the 
inverse  of  involution. 

By  Art.  27,  (2  a)»  =  4  a^;  (2  «)»  =  8  a*  ;  (2  a)*  =16  a*;  etc. 

2  a  is  called  the  secoml  or  sqiuire  root  of  4  o^  l)ecause  it  is  one 
of  the  two  ec[ual  factors  ot  4  a'^;  it  is  the  thii-d  or  cube  root  of  8  a* 
because  it  is  one  of  the  three  equal  factors  of  8a^;  etc.  Hence,  in 
general, 

A  Root  is  one  of  the  equal  factors  of  the  number  or 
expression. 

Roots  are  indicated  by  means  of  fractional  exponents, 
the  denominators  of  which  show  the  root  to  be  taken. 

Thus,  (a)*  means  the  second  or  square  root  of  a ;  (a)*  means  the 
third  or  cube  root  of  a;  (a*)*  means  the  sixth  root  of  a^.  In  general, 
(a*")*  means  the  nth  root  of  a"*. 

Roots  are  also  indicated  by  means  of  the  root  sign,  or 
radical  sign,  ^. 

Thus,  \/a  means  the  square  root  of  a  ;  ^a  means  the  cube  root 
of  a ;  v^  means  the  nth  root  of  a"». 

The  Index  is  the  number  written  ia  the  opening  of  the 
radical  sign  to  show  wliat  root  is  sought,  and  corresponds 
to  the  denominator  of  the  fractional  exponent.  When  no 
index  is  written,  the  square  root  la  understood. 


80  ELEMENTS   OF  ALGEBRA. 

« —        1 
Note,      ya  or  a»  is  defined,  when  n  is  a  positive  integer,  as  one  of  the  n 

equal  factors  of  a ;  so  that  if  Va  be  taken  n  times  as  a  factor,  the  resulting 

product  is  a ;  that  is,  (  ya)'^  or  i^a"  j"  =  a. 

,mn -\mn         (    -^\mn, 
Similarly,        (   \/a)       or  Va*""/       =  a, 

38.  The  sign,  ±  or  =p,  is  sometimes  used  and  is  called  the  double 
sign;  it  indicates  that  we  may  take  either  the  sign  +  or  the  sign  — . 
Thus,  a  ±  6  is  read  a  plus  or  minus  b. 

By  Art.  27,  (+a)4  =  a^    (-a)4  =  a4;    (+a)5  =  a5;    (-ay  =  -aK 
Therefore,   (a*)^  =  ±a;  (+  a^)^  =  a;  (—  a^)^  =  —  a.    Hence,  in 
general, 

Hven  roots  of  any  nu7nber  are  either  positive  or  negative. 

Odd  roots  of  a  nwniber  kave  the  same  sign  as  the  number 
itself. 

Since  no  even  power  of  a  number  can  be  negative,  it 
follows  that, 

An  even  root  of  a  negative  number  is  impossible. 

Such  roots  can  only  be  indicated,  and  are  called  imaginary.  Thus, 
(—  a^)^y  Y^—  6,  Y^—  1,  and  ^—  a^,  are  imaginary. 

Example  1.     Find  the  square  root  of  9  a^b^c^. 

Solution.  Since,  to  square  a  monomial,  we  multiply  the  expo- 
.nent  of  each  factor  by  2,  to  extract  the  square  root  we  must  divide 
the  exponent  of  each  factor  by  2.  The  two  equal  factors  of  9  are 
3  X  3,  or  32.  Dividing  the  exponent  of  each  factor  by  2,  we  have 
3  a^  b'^  c.  Since  the  even  root  of  a  positive  number  is  either  positive 
or  negative,  the  sign  of  the  root  is  either  plus  or  minus. 
.-.    y/9aH^  =  ±3a^b^c. 

Example  2.     Find  the  fifth  root  of  -  32  a^^  x"*. 

Solution.  Since,  to  raise  a  monomial  to  the  fifth  power,  we  mul- 
tiply the  exponent  of  each  factor  by  5,  to  extract  the  fifth  root  we 
must  divide  the  exponent  of  each  factor  by  5.  The  equal  factors  of 
32  are  2  X  2  X  2  X  2  X  2,  or  2^.     Dividing  the  exponent  of  each 


EVOLUTION.  81 

factor  by  6,  we  have  2  a*  x*.     Since  the  odd  roots  of  a  number  have 
the  same  sign  a.s  the  number  itself,  the  sign  of  the  rout  is  minus. 

.♦.  J^—  32  a*®  x*»  =  —  2  a*  a*.    Hence,  in  general, 

To  find  any  Boot  of  a  Monomial  Resolve  ike  numerical 
coejffkient  into  its  prime  factors,  each  factor  being  written 
with  its  highest  exponent,  divide  the  exponent  of  each  factor 
by  the  index  of  the  required  root,  and  take  the  ^product  of 
the  resulting  factors.  Give  to  every  even  root  of  a  positive 
expression  tlie  sign  ± ,  and  to  every  odd  root  of  any  expres- 
sion the  sign  of  the  expression  itself 


Hote.    Any  root  of  a  fraction  is  found  by  taking  the  required  root  of  each 
V27 


of  ito  terms.    Thus,   .Vl  _  J^  =  2      in  general,  t/- = -^ 


Exercise  31. 
Find  the  value  of  the  following  expressions : 

1.  V25^;  ^-%aH^a^',  y/-12baH^',  v^81ai«W 

2.  (-343a*5-6)i;  {\{7^ifz^)^;  {-x^^y^^)^-  v'Sl^V^. 

3.  v^iw.  (I21a:i2y2)i.    V25  aH"^-,    (16a-8&8)i 

4.  (-  243  a^*  Jio«)i.    (_  54  ,,,3  ^6  ^^J.    (^w  ^80)tV. 

5.  (-32ai0y-5)i;  V^2W^r^ -,   (625  a^  6i«  c*)i. 

6.  (512an8ci5rf-8)i;  V64a-«&"*;  -y/m ;  ^/-  32  ai^. 

8.    V'i^^;  (2"a2«54n2^)i;  V8l2:5i-y2m+4.  >^_  8  a;8*-«ye-+». 

A 


82  *  ELEMENTS   OF  ALGEBRA. 

10.  \/l6x^''y^''z^;  (-  iV^  ^^- ^  ?i" ^M ;  V^-^^Sm^-m 
Simplify : 

11.  Sj^a^hU-l  +  iij  a^  hh  d)^  -  (f  i  a*  ol  c-^)i 

—  V  ^^  <^^«  c    4. 
V   50 

Express  the  nth  roots  of: 

12.  3x7x4;  52:"2/2^  3a^&3.  (a-ir)3;  (^^z)";  ^'"-y"; 

Express  by  means  of  exponents : 

13.  \/JWc^;  7a(x-yT;  V^^^;   V^(^  +  2/)~ 

Queries.  If  n  and  j9  in  the  last  two  parts  of  Ex.  13  are  integral, 
what  signs  should  the  roots  have  ?  Why  ?  When  should  the  first 
two  roots  have  the  double  sign  ? 

39.    By  Art.  28,      (a  +  hy  =  a^  +  2ah -\- b^. 

Therefore,  (a^  +  2  a  6  +  62)i  =  a-^b. 

By  observing  the  manner  in  which  a  +  b  may  be  obtained  from 
a^  +  2ab  +  b^,  we  shall  be  led  to  a  general  method  for  finding  the 
square  root  of  any  polynomial. 

Process.  a"^  +  2ab -h  b^(a  +  b 

First  term  of  the  root  squared,  a^ 

First  remainder,  2  ab  +  b^ 

Trial  divisor,  2  a 

Complete  divisor,         2a  +  b 


Complete  divisor  X  b,  2ab  +  b^ 

Explanation.     The  square  root  of  the  first  term  is  a,  which  is 

the  first  term  of  the  required  root.  Subtracting  its  square  from  the 

given  expression,  the  remainder  is  2ab  +  b^,  or  b  times  2  a -\- b. 


EVOLUTION.  83 

Since  the  first  terra  of  the  i-einainder  is  twice  the  product  of  the  fii-st 
and  last  terras  of  the  root,  and  we  have  found  the  first  terra  ;  there- 
fore, divide  2  a6  by  twice  the  fii-st  terra  of  the  root  already  found,  or 
2  a.  'I'he  result  will  be  the  second  terra  b  of  the  required  root. 
Adding  6  to  the  trial  divisor  gives  the  complete  divisor,  2  a  +  6. 
Multiplying  by  6  and  subtracting,  there  is  no  remainder. 

By  Art  28,  (a-\-b+cy  =  a^+2ab-{-b^+2ac  +  2bc  +  c^. 

Therefore,  (a^-{-2ab-^b^-\-2ac-^2bc  +  c^^  =  a  +  b+c. 

Process.  a^-\-2ab-\-b^+2ac-^2bc  +  c'^{a-\-b  +  c 

First  terra  of  root  squared,  a^         

First  remainder,  2ab  +  b^-{-2ac  +  2bc+c^ 

First  trial  divisor,  2  a 

First  complete  divisor,  2a-\-b 


First  complete  divisor  X  6,        2ab-{-b^ 

Second  remainder,  2ac-\-2bc-^c* 

Second  trial  divisor,  2a-|-26        I 

Second  complete  divisor,  2a-\-2b-i-c  \ 

Second  complete  divisor  X  c,  2ac+2bc-\-c^ 

Xtxplanation.  Proceeding  as  before,  the  first  two  terms  of  the 
root  are  found  to  be  a  +  6.  To  find  the  last  terra  of  the  root,  take 
twice  the  terms  of  the  root  already  found  for  the  second  trial  divisor. 
Dividing  2  a  c  by  the  first  terra,  the  result  c  will  be  the  third  term  of 
the  required  root.  Adding  this  to  the  trial  divisor,  gives  the  entire 
divi.<*or.  Multiplying  by  c  and  subtracting  there  is  no  remainder. 
We  have  actually  squared  the  root  and  subtracted  the  square  from 
the  given  expression.     Hence,  in  general, 

To  find  the  Square  Root  of  any  Polynomial  Arrange  the 
terms  according  to  the  powers  of  one  letter.  Find  the  square  root 
of  the  first  term.  This  will  be  the  first  term  of  the  required  root. 
Subtract  its  sciuare  from  the  given  expression.  Divide  the  first  term 
of  the  remainder  by  twice  the  root  already  found.  The  quotient 
will  be  the  next  term  of  the  root.  Add  the  quotient  to  the  divisor. 
Multiply  the  complete  divisor  by  this  terra  of  the  root,  and  subtract 
the  product  frora  the  remainder.  For  the  next  trial  divisor,  take 
two  times  the  terms  of  the  root  already  found.  Continue  in  this 
manner  until  there  i-*  no  remainder. 


84 


ELEMENTS   OF  ALGEBRA. 


Example.     Find  the  square  root  of  4  a  —  10  a^  +  a^  +  4  a^  +  1. 
Arranging  according  to  the  ascending  powers  of  a,  we  have, 

[_2a2-a3 


Process. 

First  term  of  root  squared, 
First  remainder, 
First  trial  divisor,  2 

First  complete  divisoi',  2+2a 


2  a  times  first  complete  divisor, 
Second  remainder, 
Second  trial  divisor,         2+4  a 
Second  complete  divisor,  2+4  a— 2  a^ 


1+4  a  -10a3+4a5+a6(i4-2a 

_1 

4a-10a3+4a5+a6 


4a^+4q 

-4a'-^-10a3+4a5+a« 


—  2a^  times  second  complete  divisor, 
Third  remainder, 

Third  trial  divisor,  2+4  a— 4  a^ 

Third  complete  divisor,  2+4  a— 4  a^—cfi 


-4a2-8a8+4a^ 
-2a8-4a4+4a6-(-a« 


a*  times  third  complete  divisor. 


~2a3-4a4+4a5+a« 


Note.  The  student  should  notice  that  the  sum  of  the  several  subtrahends 
is  the  square  of  the  root,  and  that  he  has  actually  squared  the  root  and  sub- 
tracted the  square  from  the  given  expression. 

Exercise  32. 

Find  the  square  roots  of: 

1.  2^-4^3+ (]y2_4y4.i.  9^4- 12 a3_ 2^2+4^  +  1. 

2.  4a6_  l2a^l-~  11  a^^ly^  +  b^  a^ h^ -  17 aH^- 70 ab^ 
+  49  &6. 

3.  x^  -  12  a;5  +  60  x^  -  160  a^  +  240  x'^  -  192  a;  +  64. 

4.  8  a  +  4  +  a^  +  4  ^3  +-  8  a2 

5.  9  +-  a;6  4-  30  :r  -  4  a^  +  13  x^  +  Ux^  -  Ux^, 

6.  6a62c_4a2&c  +  ^252  +  4ft2c2  +  9  52^2  -  12  a.&c2 

7.  49  a^  -  28  a^  -  17  a^  +  6  «  4-  f. 


EVOLUTION.  85 

8.  4x^  +  9  y^  +  25  cc^^  12x1/ -SO  ay- 20  ax, 

9.  VL^  —  6  a  771^  +  15  a2  7n*  —  20  a^  vi^  +  15  a* m^  —  6a^7n 

10.  1  -2  a  +  3  a2  -  4  a3  4-  5  rt*  -  4  a5+  3  a«—  2  a^  +  a^. 

11.  9  vi^  —  G  7«  n  +  30mx+6my-\-7i^—i0nx—2  ny 
+  25r»+  lUa;^+  //2. 

12.  7^-\-l5x^{/^^-  15aj*//2  4.  ,/6_  0  ^y  _  20xV-  <^^"^y- 

13.  49x^/- 24x^/3 -30  ar8y  + 25  a^+  163^. 

14.  a;6  -  G  2:^  +  172:4  -  34  2:3  _^  40  a:2  _  40  2:  +  35. 

15.  4  -  IG  «t  +  IG  a^  +  12  rf  -  24  at  ?>i  +  9  h. 

10.  ^a:4_|^y^_^,,2,,2_,^y^^,^4.  2:4«_  0a;3n+ 5^* 
4-  12  a;"  +  4. 

17.  25  ^1  +  1 G  -  30  X  -24:xh  +  49  .rf 

18.  9x-2+ 122;-V^-6a;  +  42^-4ar^7/^  +  ^. 

40.  Since  the  square  root  of  an  expression  is  either  +  or  — ,  the 
square  root  o[  a^  -\-  '2  a  h  -\-  h^  is  either  a  +  h  or  —  a  —  h.  In  the 
process  of  finding  the  sfjuare  root  of  a'*  +  2  a  6  +  />',  we  herein  by  tak- 
ing the  square  root  of  a*,  and  this  is  either  +  a  or  —  a.  If  we  take 
—  a,  and  continue  the  work  as  in  Art.  39,  we  get  for  the  root  -^a-^h. 
Also,  the  square  root  of  a'  —  2  a  6  4-  />^  is  either  a  —  h  ov  —  a  -{-  b. 
This  is  true  for  every  even  root.  Hence,  the  signs  of  all  the  terms  oj 
an  even  root  may  he  changed^  and  the  number  will  still  be  the  root  oj  the 
same  expression.  Thus,  last  process  Art.  39,  if  —  1  he  taken  for 
the  square  root  of  1  we  shall  arrive  at  the  result  —  1  —  2  a  +  2  a'*  4-  a*. 

41.  Square  Root  of  Numerical  Numbers.  The  method 
for  extracting  the  square  root  of  arithmetical  numbers  is 
based  upon  the  algebraic  method. 


86  ELEMENTS  OF  ALGEBRA. 

Since  the  square  root  of  100  is  10,  of  10000  is  100,  etc.,  it  fol- 
lows that  the  integral  part  of  the  square  root  of  numbers  less  than  100 
has  one  figure,  of  numbers  between  100  and  10000  two  figures,  and  so 
on.     Hence, 

If  a  point  he  placed  over  every  second  figure  in  any  numher,  begin- 
ning with  units'  place,  the  numher  oj  points  ivill  show  the  numher  of 
figures  in  the  square  root. 

Thus,  the  square  root  of  324947  has  three  figures  ;  the  square  root 
of  441  has  two  figures.  If  the  given  number  contains  decimals,  the 
number  of  decimal  places  in  the  square  root  will  be  one  half  as  many 
as  in  the  given  number  itself.  Thus,  if  2.39  be  the  square  root,  the 
number  will  be  5.Vi2i;  if  .239  be  the  root,  the  number  will  be 
6.057121 ;  if  10.321  be  the  root,  the  number  will  be  106.523041. 
Hence, 

The  numher  of  points  to  the  left  of  the  decimal  point  will  shorn  the 
numher  of  integral  places  in  the  root,  and  the  numher  of  points  to  the 
right  will  show  the  numher  of  decimal  places. 

Example  1.    Find  the  square  root  of  45796. 

a  +&+c  =  214 
Process.  45796(200+10+4  =  214 

The  square  of  a  or  200,  -  40000 

First  remainder,  5796 

First  trial  divisor,  2  a,  or  400        I 

First  complete  divisor,  2a-lh,  or  410  [ 
First  complete  divisor  X  &,  or  10,  4100 

Second  remainder,  1696 

Second  trial  divisor,  2a+2&,  or  420  i 

Second  complete  divisor,  2a+2&+c,  or  424  | 
Second  complete  divisor  X  c,  or  4,  1696 

Explanation.  There  will  be  three  figures  in  the  root.  Let 
a  -^h  -\-  c  denote  the  root,  a  being  the  value  of  the  number  in  the 
hundreds'  place,  h  of  that  in  the  tens'  place,  and  c  the  number  in  the 
units'  place. 

Then  a  must  be  the  greatest  multiple  of  100  whose  square  is  less 
than  45796,  this  is  200.  Subtract  a^,  or  the  square  of  200  from  the 
given  number.     Dividing  the  first  remainder  by  2  a,  or  400,  gives  10 


EVOLUTION. 


87 


for  the  value  of  b.  Add  this  to  400,  multiply  the  result  by  10  and 
subtract.  Dividing  the  second  remainder  by  2  a  -f  2  6,  or  420,  gives 
4  for  the  value  of  c.  Adding  this  to  420,  multiplying  and  subtract- 
ing, there  is  no  remainder.  Hence,  214  is  the  required  root;  because 
we  have  actually  squared  it  and  subtracted  this  square  from  the 
given  number  and  found  no  remainder.  The  student  should  observe 
that  the  .sum  of  the  several  subtrahends  is  the  square  of  the  root. 
Example  2.     Find  the  square  root  of  17.3  to  lour  decimal  places. 


Process. 

S<iuare  of  4, 

First  remainder, 

First  trial  divisor,  8 

First  complete  divisor,  81 

First  complete  divisor  multiplied  by  1, 

Second  remainder. 

Second  trial  divisor,  82 

Second  complete  divisor,  825 


17.360<KK)06(4.1693..., 

16 

130 


81 
4900 


Second  complete  divisor  nmltiplied  by  5, 

Third  remainder, 

Third  trial  divisor,  830 

Third  complete  divisor,  8309 


Third  complete  divisor  multiplied  by  9, 
Fourth  remainder. 
Fourth  trial  divisor,  8318    I 

Fourth  complete  divisor,  83183    | 
Fourth  complete  divisor  multiplied  by  3, 
Fifth  remainder, 


4125 
77500 


74781 
271900 


249549 
22351 


Let  the  student  formulate  a  method  for  arithmetical  square  root 
from  what  has  been  demonstrated. 

NotM ;  1.  If  the  trial  divisor  is  not  contained  in  the  remainder,  annex  0  to 
the  root,  also  to  the  divisor,  then  annex  the  next  period  and  divide. 

2.  Should  it  be  found  that  after  completing  the  trial  divisor,  it  gives  a  pro- 
duct greater  than  the  remainder,  the  quotient  is  too  large,  and  a  less  quotient 
must  be  taken. 

3.  Tf  the  last  remainder  is  not  a  perfect  square,  annex  periods  of  ciphers  and 
procee<l  as  before. 

4.  The  square  root  of  a  fraction  may  be  found  by  taking  the  square  root  of 
its  terms,  or  by  first  reducing  it  to  a  decimal. 


88  ELEMENTS   OF  ALGEBRA. 

Exercise  33. 

Find  the  square  roots  of : 

1.  33124;  41.2164;   Jf | ;    ^%^^^;   .099225;    1.170724. 

2.  .30858025;  5687573056;  943042681. 
Find  the  square  root  to  four  decimal  places  of: 

3.  .081;  .9;  .001;  .144;  if;  .00028561;  3.25;  20.911. 

42.    By  Art.  29,  (a  +  by  =  a^  +  3  a^ b  +  3 ab^  +  bK 

Therefore,    (a^  +  3  a^  6  +  3  a  &2  +  b^)l  =  a  +  b. 

By  observing  the  manner  in  which  a  -\-b  may  be  obtained  from 
a^  +  3  a^ft  +  3 a  &2  -I-  6"^,  we  shall  be  led  to  a  general  method  for  find- 
ing the  cube  root  of  any  compound  expression. 

Process.  a^+2a%yMb'^-\-b^  {a+b 

First  term  of  the  root  cubed,  o^ 

First  remainder,  3a%+Zab'^-{-b^ 

Trial  divisor,  or  3  times  the  square  of  a,   Za^ 

3  times  the  product  of  a  and  b,  2ab 

Second  term  of  the  root  squared, 6^ 


Complete  divisor,  Za^+Zab+b'^ 

Complete  divisor  X  &,  Za%+Zab'^^-b^ 

Explanation.  The  cube  root  of  the  first  term  is  a,  which  is  the 
first  term  of  the  required  root.  Subtracting  its  cube  from  the  given 
expression,  the  remainder  hZaH  +  Zab'^+b'^,  or  b  times  Za^  +  Zab+b^ 
Since  the  first  term  of  the  remainder  is  three  times  the  product  of  the 
square  of  the  first  term  of  the  root  multiplied  by  the  last  term,  divide 
Za^b  by  three  times  the  square  of  the  first  term  of  the  root  already 
found.  The  result  will  be  the  second  term  6  of  the  required  root. 
Adding  to  the  trial  divisor  three  times  the  product  of  the  first  and 
second  terms  of  the  root,  and  the  square  of  the  second  term,  gives  the 
complete  divisor,  ov  Zw^^  Zab  +  b^.  Multiplying  by  b  and  subtract- 
ing, there  is  no  remainder. 

Since  the  cube  of  a+b-\-c  is  a^-\-Za'^b^-Zab'^-\-h^+Za^c  +  Qabc 
-f-3&2c  +  3ac2+36cHc8,  the  cube  root  of  a8  + 30^6 -|- 3 a 62+63+ 3 a^c 
+  6a6c  +  362c+3acH36c2+c«  is  a  +  b  +  c. 


EVOLUTION. 


89 


i 

CO 


4- 

^ 

^j 

o 

eo 

cS 

j; 

w 

^ 

1 

t 

s 

5 

-- 

^ 

^ 

^ 

t 

^ 

1 

•o 

e^ 

« 

$ 

# 

c    o 


J 


cS 


<Sc^'" 


^5  ^ 


o 

I 


«  ^        X 


^ 


iz 


•5    S  ,2 

O      C  ;:2 


'tis       -k.) 


I- 

Ph    C 


■2  5 


-^  -r  -^  -«  -^  •> 

g    ^  3  B  .5  -^ 

"^    q      O)      4>  eS  ^_^ 

•2  iJ  s  S  S  is  Ts 

£  -3  8  8  -2  '^  § 

£   I  ^  ^  §  §^ 


3,3. 

s  a 

8  8 

o    p 


90  ELEMENTS  OF  ALGEBRA. 

Explanation.  Proceeding  as  before,  the  first  two  terms  of  the 
root  are  found  to  be  a  +  h.  To  find  the  last  term  of  the  root,  take 
three  times  the  square  of  the  terms  of  the  root  already  found  for  the 
second  trial  divisor,  and  divide  Za^c  by  the  first  term.  The  result 
will  be  the  third  term  of  the  required  root.  Adding  to  the  second 
trial  divisor  three  times  the  product  of  a  +  &  and  c,  and  the  square  of 
c,  gives  the  second  complete  divisor.  Multiplying  by  c  and  subtract- 
ing, there  is  no  remainder.  Observe  that  the  sum  of  the  several  sub- 
trahends is  the  cube  of  the  root,  and  that  we  have  actually  cubed  the 
root  and  subtracted  the  cube  from  the  given  expression.  Hence,  in 
general, 

To  find  the  Cube  Root  of  any  Polynomial.  Arrange  the  terms 
according  to  the  powers  of  one  letter.  Find  the  cube  root  of  the  first 
term.  This  will  be  the  first  term  of  the  required  root.  Subtract 
its  cube  from  the  given  expression.  Divide  the  first  term  of  the 
remainder  by  three  times  the  square  of  the  root  already  found.  The 
quotient  will  be  the  next  term  of  the  root.  Add  to  the  trial  divisor 
three  times  the  product  of  the  first  and  second  terms  of  the  root,  and 
the  square  of  the  second  term.  Multiply  the  complete  divisor  by 
this  term  of  the  root,  and  subtract  the  product  from  the  remainder. 
For  the  next  trial  divisor,  take  three  times  the  square  of  the  root 
already  found.  Continue  in  this  manner  until  there  is  no  remainder 
or  an  approximate  root  found. 

A  Term  may  be  a  figure,  or  a  letter,  or  a  combination  of 
figures  and  letters,  or  of  letters  only,  produced  by  multi- 
plication or  division,  or  both. 

Thus,  in  the  algebraic  expression  5  +  2a®6*  — a+  -gi^;  5,  2a^b*,  a, 
ah-  "^y 


«2  ,,w 


are  terms. 


An  Algebraic  Expression  is  a  representation  of  a  number 
by  any  combination  of  algebraic  symbols. 

Example.     Find  the  cube  root  of  27  a  — 8  a^- 36+ 36 a^- 12a- ^ 
-54a^  +  9a-§+27a§  +  a-6-6a-J. 

The  work  is  conveniently  arranged  as  follows  : 


EVOLUTION. 


91 


+ 

I 

1 

+ 

+ 

•4M 

•* 

HB« 

o 

l„ 

1 

<N 

C3 

o 

J 

CO 

«o 

CO 

1 
1 

1 

1 

**— ^ 

^ 

e 

C5 

05 

4- 

+ 

r* 

k 

1 

I 

1— • 

1 

CO 

1 

1 

o 

1 

t^ 

r^ 

04 

(N 

+ 

+ 

■an 

im 

c2 

1 

<2 

1 

1 

1 

5 

^ 

CO 

CO 

+ 

+ 

»«• 

HM 

1 

1 

1 

1 

1 

C      8 

t-     I- 

<M     CN 

+ 

Mi 

CD 
I 

I 

4- 


4,      ♦-      j-J       4,       O) 

02    CO    H    cc   cw 


92  ELEMENTS  OF  ALGEBRA. 

Exercise  34. 

Find  the  cube  roots  of : 

1.  x^—  Sx^  +  5x^-  3x  --1;  x^-  Saa^+  5  a^ a^ 

—  S  a^  X  —  a^. 

2.  8  0^6  +  48  ax^+60  a^x'^-80  a^x^-  90  a^x^+lOSa^x 

-  27  a^. 

3.  x^-6x^+  15  x^-20  a^+15x^-6x+l. 

4.  27a^-54.a^h  +  9a^^+2SaH^-3a^b^-6ah^-h^ 

5.  Sx^+  12  x^  -SOx*-  35  x^  +  4:5x^  +  27  x  -  27, 

6.  216  +  3422:2+i7i^4  4.27^6_27^5_i09^-108ic. 

7.  a3  -  3^25  -  53+  8c3+  6a2c-12a&c  +  6fe2c 
4-12ac2- 125c2+  3ah\ 

8.  1  -  3rK  +  62^2  -  10  rt3  +  12  ic^  -  12  a;^  +  10  ri;^  -  6  a;^ 
4-  3  a:-8  -  .^'9. 

9.  8  x^  -  36  x^y  +  114  aj*?/^  _  207  x^y^  +  285  0^23^ 
-225  0^2/5+125  7/6. 

10.  a^  +  6a^h  -  Sa^c  +  12  a V^  -  12  abc  +  3ac2 
+  8^3-  12h^c+  6bc^-c^. 

11.  x^  +  Sx^y-Sa^y^-~lla^f+6xh/^+12xf  -  8/. 

12.  204  rr*2/2  - 144  a;^ 7/  +  8  /-  36  o;^/^  _  171  ^^s^^s  _|.  54^6 
+  102  0^2/*. 

43.  Cube  Root  of  Numerical  Numbers.  The  method  for 
extracting  the  cube  root  of  arithmetical  numbers  is  based 
upon  the  algebraic  method. 


EVOLUTION.  93 

Since  the  cube  root  of  1000  is  10  ;  of  1000000  is  100,  etc.,  it  fol- 
lows that  the  integral  part  of  the  cube  root  of  numbers  less  than  KXX) 
has  one  figure,  of  nuiubers  between  1000  and  1000000  two  figures, 
and  so  on.     Hence, 

I/a  point  be  placed  over  every  third  fgure  in  any  number,  beginning 
with  units'  place y  the  number  of  points  will  show  the  number  of  figures 
in  the  cube  root. 

Thus,  the  cube  root  of  274625  has  two  figures ;  the  cube  root  of 
109215352  has  three  figures. 

If  the  given  number  contains  decimals,  the  number  of  decimal 
places  in  the  cube  root  will  be  one  third  as  many  as  in  the  given 
number  itself.  Thus,  if  1,11  be  the  cube  root,  the  number  will  be 
1.367631 ;  if  .111  be  the  root,  the  number  will  be  (3.00i36763i  ;  if 
11.111  be  the  root,  the  number  will  be  1371.706960631.     Hence, 

The  number  of  points  to  the  left  of  the  decimal  point  will  show  the 
number  of  integral  places  in  the  root,  and  the  number  of  points  to  the 
right  will  show  the  number  of  decimal  places. 

Example  1 .    Find  the  cube  root  of  778688. 

a  +  6  =  92 

Process.  778688  (  90  +  2  -=  92 

The  cube  of  a,  or  90,  729000 

First  remainder,  49688 

First  trial  divisor  3  a",  or  3  (90)^  =  24300 

3  times  the  protluctof  a  and  6,  or  3X90X2=     540 
Second  term  b  of  the  root  squared,  2*'*        = 


First  complete  divisor,  24844 

First  complete  divisor  X  6,  or  2,  49688 

Explanation.  There  will  be  two  ^gures  in  the  root.  Let  a  -f-  6 
denote  the  root,  a  being  the  value  of  the  number  in  tens*  place,  and 
b  the  number  in  units'  place.  Then  a  must  be  the  greatest  nmltiple 
of  10  whose  cube  is  less  than  778688,  this  is  90..  Subtract  a«,  or 
the  cu>)e  of  90,  from  the  given  number.  Dividing  the  remainder 
by  3  a*,  or  24.3()0,  gives  2  for  the  value  of  6.  Add  to  the  trial  divisor 
3  a  6,  or  54(),  and  6^,  or  4,  for  the  complete  divisor.  Multiplying  by 
2  and  subtracting,  there  is  no  remainder.     Hence,  92  is  the  required 


94 


ELEMENTS  OF  ALGEBRA. 


root,  because  we  have  actually  cubed  it  and  subtracted  this  cube  from 
the  given  number  and  found  no  remainder. 

Example  2.     Find  the  cube  root  of  897.236011125. 


=  24300 

1620 

36 


25956 


Process. 

Cube  of  9, 

First  remainder, 

First  trial  divisor,  3  times  (90) ^ 

3  times  the  product  of  90  and  6, 

6  squared, 

First  complete  divisor, 

First  complete  divisor  multiplied  by  6, 

Second  remainder. 

Second  trial  divisor,  3  times  (960)2  -  2764800 

3  times  the  product  of  960  and  4,  11520 

4  squared,  16 
Second  complete  divisor,                        2776336 
Second  complete  divisor  multiplied  by  4, 
Third  remainder, 

Third  trial  divisor,  3  times  (9640)2  =  278788800 
3  times  the  product  of  9640  and  5,  144600 

6  squared,  25 

Third  complete  divisor,  278933425 

Third  complete  divisor  multiplied  by  5, 


897.236011125(9.645 

729 

168236 


155736 
12500011 


11105344 
1394667125 


1394667125 


Let  the  student  formulate  a  method  for  arithmetical  cube  root  from 
what  has  been  demonstrated. 

Note.     The  notes  in  Art.  41  are  equally  applicable  to  cube  root,  except  that 
in  Note  1  two  ciphers  must  be  annexed  to  the  divisor  instead  of  one. 


Exercise  35. 

Find  the  cube  roots  of: 

1.  74088;  34012.224;  .244140626. 

2.  ^mif^;  .000152273304. 


EVOLUTION.  95 

Find  to  three  places  of  decimals  the  cube  roots  of : 
3.    .64;  .08;  8.21;  .3;  .008;  J;  ^. 

44.    Since        a*  =  c^^  =  (a*)*  =  Vo*  =  VVa, 

The  fourth  root  is  the  square  root  of  the  square  root. 

* 

Since  a*  =  a^  '<3  =  (aSf  =  Va*  =  V  4^, 

The  sixth  root  is  the  cube  root  of  the  square  root.     Hence, 

When  the  root  indices  are  composed  of  factors,  the  ope- 
ration is  performed  by  successive  extraction  of  simpler 
roots. 

Hote.  It  is  suggested  that  the  teacher  use  the  remainder  of  this  article  at 
his  discretion. 

We  may  find  the  fifth,  seventh,  eleventh,  or  any  root  of  an 
expression  or  arithmetical  number  if  desired,  by  using  the 
form  for  completing  the  divisor.     Thus, 

To  find  the  fifth  root. 

Form,  (a  +  6)»  =  a*  +  (5  a*  +  10  a«6  +  10  a*^**  -j-  5  a  6»  +  6*)  b. 

Trial  divisor,  5  a*. 

Complete  divisor,  (5  a*  -h  10  a»  6  +  10  a*  &«  +  5  a  6»  +  b*) . 

To  find  the  seventh  root. 

Form,  (a+6)'  =  a7+(7a»+21a»H35a<6«-f35a»6»+21a26<+7a6»+6«)6. 

Trial  divisor,  7  a*. 

Complete  divisor,   {7a*+2la^b+35a*b^+3ba%^-2la^b^-7ab^+h% 


96 


ELEMENTS  OF  ALGEBRA. 


Example.     Find  the  fifth  root  of  36936242722357. 


Process. 

36936242722357(517 

a5  =  5S  = 

3125 

First  remainder, 

56862427 

First  trial  divisor  =:5a'^(a  considered 

as 

5  tens)            =  5  (50)*      = 

31250000 

10 a^b  (h  considered  as  1  unit)  = 

10(50)3  X    1             ::= 

1250000 

10a2  62=  10  X  (50)-^X  (1)^    = 

25000 

5  a  63     =    5X  (50)  X  (l)^  ^= 

250 

6*           =  (1)^ 

1 

First  complete  divisor, 

32525251 

First  complete  divisor  multiplied  by 

1,                 32525251 

Second  remainder. 

2433717622357 

Second  trial  divisor  =  5  a^  (a  considered 

as  51  tens)  =  5  X  (510)^  =     338260050000 
10  a^b{b  considered  as  7  units) 

=  10  X  (510)3  X  (7)  = 
lOa-^62  =  10  X  (510)2  X  (7)2  = 
5a68  =  5X(510)  X  (7)3  = 
6*  =  (7)4 

Second  complete  divisor,  347673946051 

Second  complete  divisor  multiplied  by  7,  2433717622357 


9285570000 

127449000 

874650 

2401 


Miscellaneous  Exercise  36. 

Express  the  nth.  roots  of: 

1.  ah^c-'^;  52-a;3"(^-;y  +  2«)4"x2%^4-7/"y"x4"(a;— 2/T- 

2.  Simplify  4:  a  (Sax  y)^  —  5  xhj^  ^2S^  «^  X  ak 
Find  the  square  roots  of: 

3.  ^x  +  l^-\-x-^—4x-^^-l2x^. 

4.  28-24tt-^-16al  +  9ft-"  + 4a^ 


EVOLUTION.  97 

5.  162;'^"+16ic7--4a;8*-4x-9''  +  a:io». 

6.  a^u~l—4:xhj~^  +  6  —  4  x~^  yi  +  x~^t/^. 

7.  6«cr/;5  +  4  62u;*  +  a^x^^  -^  \)  c^ -12  bcx^  -  4:  abx'. 

8.  {x'^  +  Ax^  +  ^ax^  -^  ia^-2x^-^  ax. 

9.  a2»+  2a\6"'+  z'"";  «  ±  2  ai  a;i  +  x. 
Find  the  cube  roots  of : 

10.  60  x^i/ -^  4S xf  ^27 x^  +  lOS x^i/ -90 si^y^ -{-  8/ 
-80a^^. 

11.  24a;*'" 2/2-  4-  %  3^'^yin_  6a^'»7/"  +  a:6m_9g^^n 
+  64/"-  56a;3«2^* 

12.  15x-*  -  6:r-i—  62;-6  +  15aj-2+  1  +  a:-6-^20a;-8. 

13.  Su^-^2^f  +  ixi/-^f. 

14.  ^a-i-6a-J-J  +  8a-t-^a-8  +  27a-i  +  54a-4 
+  |a-t  +  36  a-4- 18a-2. 

Find  the  sixth  roots  of: 

15.  1215a*-1458a6+135a2-540a3_l8«  +  l+729a« 

16.  .x^+  f-6x  ?/+  15jy^ij^-Gx^y+  15 3^1/- 20 a^y^l 

17.  160  a3  +  240  a*  +  60  a2  +  192  a^  +  64  a«+  12  a  +  1. 

18.  2985984;  262144. 
Find  the  eighth  roots  of : 

19.  </8+28a3+8a+l  +  56a8+70a*  +  8aH56«6^28a«. 

20.  («*  +  5*  -  2  a&8  +  3  a2i^  -  2  aH)*. 

7 


98  ELEMENTS   OF   ALGEBRA.  -^ 

21.  Find  the  5tli  root  of  36936242722357. 

22.  Find  the  7th  root  of  1231171548132409344. 
Extract  the  following  roots  : 

23.  y/(a4  +  19  a2  +  25  -  6  aS  _  30  a), 

24.  y/[^^  —  2  {m  +  71)0^  +  (m^  +  4:mn  -\-  n^)  x^ 

—  2mn  (m  -\-  n)  x  +  7/^^  71^] 2 

25.  [25  a2  _  20  a  5  +  4  52  +  9  c2  -  12  6c  +  30  ac]i 

26.  [27a^- 54a5+  63  a^-Ua^^  21a2_6a  +  l]i 

27.  y/(aj2'"  +  2  af"-^'^  -  2  ^'"^^  +  ^2»  _  2  aj"+^  +  ^). 

28.  [a6_  12^5+  60^4-  160^3+  240  a^-192a+ 64:]^. 

29.  [(«4-fe)6'»^34.6^n^(^^j^4m^_^^2rt2V(a+&)2'"ic+8a3V]i 

30.  y/(^2«  ^  2  a:2«-l  +  3  3^2—2  _^   2  a;2'»-3  +  aj2n-4)^ 

31.  y^(8  -  12a3«-i  +  6a6"-2_a9«-3^^ 

Queries.  What  signs  are  given  to  even  and  odd  roots  ?  Why  ? 
What  principles  govern  the  signs  of  roots  ?  Upon  what  principle  is 
the  method  lor  finding  the  root  of  a  monomial  hased  1  How  derive 
the  method  for  finding  the  square  root  of  any  polynomial '?  Why 
divide  the  first  term  of  the  remainder  hy  twice  the  terms  of  the  root 
already  found  for  the  next  term  of  the  root  ?  Why  add  the  quo- 
tient to  the  trial  divisor  for  the  complete  divisor  ?  How  derive  the 
method  for  finding  the  cuhe  root  of  any  polynomial  ?  Why  divide 
the  first  term  of  the  remainder  by  three  times  the  square  of  the  root 
already  found  for  the  next  term  of  the  root  ?  Why  add  to  the  trial 
divisor  three  times  the  product  of  the  terms  of  the  root  already  found 
by  the  next  term,  and  the  square  of  the  next  term,  for  the  complete 
divisor  ? 


USE   OF   ALGEBRAIC  S\MBULS.  99 


CHAPTER   VIII. 
USE  OF  ALGEBRAIC   SYMBOLS. 

45.  Symbols  of  operation  are  used  to  indicate  that 
algebraic  operations  are  to  be  performed. 

Thus,  m  -f  (a  -  6)  indicates  that  a  —  6  is  to  be  added  to  m  ; 
m  —  (a  —  6)  indicates  that  a  —  h  is  to  be  subtracted  from  m.  Per- 
forming the  operations,  we  have, 

m-\-  (a  —  b)  =  m  +  a  —  b  ; 

m  —  (a  —  h)  =  m  —  a  +  b.     Hence, 

A  plus  sign  before  a  symbol  of  aggregation  shows  that  the  enclosed 
terms  are  to  be  added  to  what  precedes ;  as  this  operation  does  not 
change  the  signs,  the  removal  of  the  symbol  does  not  affect  the  signs. 
Removing  one  preceded  by  a  minus  sign  changes  the  sign  of  each 
enclosed  term. 


Thus,  a-26-[4a-66-{3a-c+(5a-26-3a-c  +  2  6){] 
=  a-26-[4a-66-{3a-c  +  (5a-26-3a  +  c-2  6)}] 
=  a-2  6-[4a-66-{3a-c+(2a-46  +c         )}] 

=  a-26-[4a-66-{3a-c+   2a-46  +c  |] 

=  a-26-[4a-66-{6a  -46  }] 

=  a-26-[4a-66-5a  +46  ] 

=a-26-[      -26-      o  ] 

=  a-26  +26+      a 

=  2a 

Explanation.  Remove  the  vinculum,  subtract  and  unite  like 
terras  ;  then  remove  the  parenthesis  and  unite  like  terms  ;  now 
remove  the  brace,  subtract  and  unite  like  terms  ;  finally,  removing 
the  bracket,  subtracting  and  uniting  like  terms,  we  have  2  a. 


100  ELEMENTS  OF  ALGEBRA. 

Exercise  37. 

Simplify : 


1.  2a-l3b+(2b~c)-4:c+  {2  a-(Sh-c-2h)}']. 

2.  a  —  b  +  c  -  (a  +  b  —  c)  —  (c  —  b-  a). 

3.  x^  ~[4x^~  {6  x^  -  (4.r  -  1)}]  -(x^  +  4:x^+6x^ 
H-  4^  +  1). 

4.  ~l0(x-\-y)-lz  +  x  +  y-3{x  +  2y-(z+x-y)}^ 

4-    4:Z. 


5.  a-[5b-{a-{5c-2c-b-4b)  +  2a-{a-2b-hc)}l 

6.  -5{a-6[a-{b-  c)]}  +  60  {6  -  (c  +  a)}. 
7.-2a-(3b  +  2c)-[5b-{6c-U)  +  5c-{2a-(c+2b)}]. 


8.    3:r  -  {?/  -  [?/  -  (:r  +  ?y)  -  {-  ;/  -  (^^  _  ^  -  ^z)}]}. 


9.  3a-[2b  +  a-b]-^  [Sb-  2a  +  bl 

10.  {(re-  2  ?/  +  a; ?/)  -  (a;  -  ?/  +  ^)}-{a:  -(x-2j  +  xy)}. 


11.  I  a  -  [|  a  -  {1  «  -  (2  ft  -  5  a  +  6)}  -  (f  a  -  3)]. 

12.  f{f(a-6)-8(&-c)}-{|(6-0-i(c-«)} 
-  I  {^  -  a  -  |(«  -  6)}. 

13.  5  {a  -  2  [ft  -  2  (ft  +  a;)]}  -  4  {«  -  2[a-  2  (a  +  rr)]}. 


14.  ft+25-{6a~[3&  +  (8a;-2  +  &.y-a;  +  4a)]-3/;} 
+  2(1  +  |-«-46). 

15.  2  (|&  -  f  ft)  ~  7  [ft  -  6  {2  -  5  (ft  -  b}}\ 

16.  -|{-f[-4(-:.i)]}+f{-3(-a;^)}. 

17.  -  I  {-  L"  Q^  -  h)]}  +  {-  I  [-  (ft  -  fe)]}. 


USE  OF  ALGEBRAIC   SYMBOLS.  101 

18.  5  {a  -  2[6  -  3  (c  H-  d)]}  -  4  {a  -  3  [6  -  4  (c  -  </)]}. 

19.  (a-l)(a  -  2)  -  3a(a  +  3)  +  2  {(a  +  2)(a  + 1)-  3}. 

20.  {xz  -{x'-y){y  +  z)}  -y[y-{x-  z)\ 

21.  {a^h-\-c-\-df^{a-h-c^ df+  (a-b  +  c-d)^ 
+  (a  +  b-c-  d)\ 

22.  7!'-{2xy-[-{x-{y-z}){x+{y-z})-\-2xy-\-4.yz} 

23.  a(a  +  l)(aH-2)(a  +  3)-6(2a-J)-"ta2-3a  +  l)2. 

24.  571  {(a:— y)a-&2;}— 27i{a:(a-6)-af ?/}  -  {Zax—{pzr-2x)h}  n. 

25.  (a:2  4.2^)^_(a;  +  y)(a;{7i-y}-y{^i-4). 

26.  2aVi  — 3  m  — [6yi  — 6  7i  +  (2:i-2Vi)a]  +  &X  Vy. 

27.  (9  7?l2  7l2     -     4  71*)  (7m2    -    7l2)     -     {3  77?  7?.     -     2  W^} 
{3  771  (/?t2  4-  71^)  —  2  71  (71*  +  3  771  71  —  TH?)}  71. 

28.  77l2  ( w2  +  7l2)2  -  2  7^2  ^2  (^  +  n)  (771  -  7^)  -  (tTI^ _  n3)2.. 

29.  i(^^  +  iy)(i^-Jy)-(J^-!.y)^-?(^-f2/^). 

30.    Y  a  ^  +  4  ?/')  ( J  ^  -  J  ?/')  -  (i  :r  -  3)  (J  :r  +  3) 
(4^-9)  +  (f  y  -  3)(f  y  +  3)  (^  .7/2  -  9). 

The  use  of  symbols  of  aggregation  aid  in  shortening  the 
work  in  certain  cases  in  division.     Thus, 

a  4-  (6  +  c)  )  (6  +  c)  a^  +  (62  -I-    6  c  +  c')  a  -  -  (6  +  c)  6  c  (  (6  +  c)  a  -  6c 
(6  +  c)a«  +  (62+  2  6c  +  c«)  a 


+  (    -    6c        )a-(6  +  c)6 
+  (    -    6c        )a-(6  +  c)6 


102  ELEMENTS  OF  ALGEBRA. 

Divide : 

31.  (6  +  c)a2  +  (&2  +  32^c  +  c2)a+(i  +  c)&c  by  a  +  h  +  c. 

32.  (a  -\-hf-6{a  +  b)-  27  by  (a  +  b)  +  3. 

33.  {x+ijf+  3  {x  +  yfz  +  S{x  +  y)z^+^  by  (x  +  yf  + 
2(x  +  y)z  +  z^. 

34.  (x  +  yf  _.  2  (^  +  ?/)  z  +  z^  by  x  ^  y  —  z. 

35.  {a  +  &)3  +  1  by  «  +  J  +  1. 

46.  The  converse  operation  of  enclosing  any  number 
of  terms  of  an  expression  in  a  symbol  of  aggregation  is 
important. 

a  +  m~c-\-h  —  n  =  a  +  m  —  c-\-(h  —  71). 
a  —  m  —  c  -]-  b  ~  n  =  a  —  (m  +  c)  -^  (b  —  n). 
ax^  —  ny  +  hx^  —  cy^=  {ax^  —  ny)  +  (hx^  —  c  y*). 
xy  —  ax'-by  +  ab  =  (x y  —  by)  —  (a x  —  a  b). 

Hence,  when  the  signs  +  and  —  indicate  operation  : 

(1)  Any  number  of  terms  may  be  enclosed  in  a  symbol  of  aggrega- 
tion preceded  by  the  sign  + ,  without  changing  the  sign  of  each  term. 

(2)  Any  number  of  terms  may  be  enclosed  in  a  symbol  of  aggrega- 
tion preceded  by  the  sign  — ,  if  the  sign  of  each  term  be  changed. 

The  terms  may  "be  enclosed  in  various  ways.     Thus, 

am  +  an  —  ax  —  bx  +  cy  —  dz=(am  —  ax)  +  \an  —  bx^-\-\cy  —  dz\, 
or,  am-\-an—ax  —  bx-\-cy—dz=  (am,  +  an  —  ax)—{bx  —  cy  +  dz], 
or,  am  +  an  —  ax  —  bx+cy  —  dz=  {am-\-an)—{ax-\-bx)+{cy—dz).    Etc. 

If  a  factor  is  common  to  each  term  within  a  symbol  of  aggregation, 
it  may  be  placed  outside  as  a  multiplier.     Thus, 
ax^  +  bx'^~bx^  +  dx'^={ax^--bx^)  +  {bx'^+dx^)=x^{a-b)+x'^{b  +  d). 

Note.  An  expression  consisting  of  three  or  more  terms  may  be  raised  toa 
given  power  by  inspection,  by  first  changing  it  to  the  form  of  a  binomial. 
Thus,  (a  +  &  +  c  —  (^)4  =  [(a  +  J)  +  (c  —  d)Y^  =  etc. 


USE  OF  ALGEBRAIC  SYMBOLS.  103 

Exercise  38. 

Bracket  the  last  three  terms  so  that  each  bracket  shall 
be  preceded  by  a  —  si<,'n : 

I.  a:*  -  a  x8  -  5  a^»  +  2  ;  m6  +  3  w3  +  3  -  6  m2. 

3.  4:X+'dao(^—{)7^  —  bey  -^  y;  x^—y^—z^-\-ab-\-3ac. 

4.  Express  each  of  the  above  as  binomials,  and  enclose 
the  last  two  terms  in  an  inner  brace  preceded  by  a  —  sign. 

Bracket  the  following  in  binomials,  also  in  trinomials, 
each  preceded  by  a  —  sign : 

5.  2ab  — Say-\-4:bz  — 5bx-2cd  —  3. 

6.  a— 26  +  cz  —  d  —  l-^z  —  x—2y+  2m  —  n-\-p—4ahc. 

7.  2x-Sxy-{-  4  ^f  -  5  2:3^2  +  ^,p  _  ^.y  2 

8.  a^+3  a*— 4  a^—3  CT^-f  a  —1 ;  —2  m—2>  7i+4p— 5  2;— 1—6  y. 

9.  a  n  ■\-  ab  —  a  c  —  c  X  —  a  X  —  a  y  —  3  ab  c  -\-  Z  xy  z. 

10.  Express  the  above  six  examples  in  trinomials,  and 
enclose  the  last  two  terms  in  an  inner  bracket  preceded  by 
a  —  sign. 

II.  Expand  (tw  +  2  71  —  j^. 

12.  Simplify  and  bracket  like  powers  of  x  in  2b^—ax 
-{a:i^-\bx-nx-  {a^  -^  3r.i2}-|  _  {ax'-2cx)}. 

Queries.  Why  may  a  8ymlx)l  of  aggregation  preceded  by  a  + 
sigu  be  remove<l  without  changing  the  signs  of  the  enclosed  terms  ? 
If  a  symbol  of  aggregation  pi*ecedetl  by  a  —  sign  be  removed,  why 
change  the  signs  of  the  enclosed  terms? 


104  ELEMENTS  OF  ALGEBRA. 

CHAPTEE  IX. 

SIMPLE  EQUATIONS. 

47.  3  a:  +  5  =  5  a;  —  7  is  called  an  Equation.  The  first 
memher  or  first  side  is  3^  +  5,  and  the  second  member  or 
second  side  is  5  x  —  7. 

x  =  x,  14  =  14,  are  called  Identities  or  Identical  Equations. 

To  solve  an  equation  is  to  find  the  value  of  the  unknown 
number. 

The  process  of  solving  an  equation  depends  upon  the 
following  axioms: 

1.  If  to  equal  numhers  we  add  equal  numbers,  the  sums 
are  equal. 

2.  If  from  equal  numhers  we  subtract  equal  numbers,  the 
remainders  are  equal. 

3.  If  equal  numbers  are  multiplied  by  equal  numbers,  the 
products  are  equal. 

4.  If  equal  numhers  are  divided  by  equal  numbers^  the 
quotients  are  equal. 

Example  1 .    Find  the  value  of  x  in  the  equation  6  a:— 11  —  Sx+lO. 

Solution.  Subtracting  3  x  from  each  member  of  the  equation 
(Axiom  2),  we  have  6a;  —  3a;— ll  =  3cc  —  3.X+10.  Uniting  like 
terms,  3  a;  —  11  =  10.  Adding  11  to  each  member  (Axiom  1),  and 
uniting  like  terms,  we  get  3  a;  =  21.  Dividing  both  members  by  3 
(Axiom  4)  gives  x—1. 

Proof.  To  verify  this  result,  substitute  7  for  x  in  the  given  equa- 
tion. Then,  6  X  7  -  11  =  3  X  7  +  10,  or  31  =  31,  which  is  an 
identity.     Hence,  the  value  of  x  is  7. 


SIMPLE  EQUATIONS.  105 

Example  2.  Solve  the  equation  2  (a:  -  8)  -  3  (9  -  x)  +  5  (a:  - 1 1) 
=  7  -3  (a: -17). 

Solution.  Performing  the  indicated  operations,  and  uniting  like 
terms,  10  a;  —  98  =  58  —  3  z.  Adding  3  x  and  98  to  each  member  of 
the  equation,  we  have  10  «  +  3  a:  -  98  +  98  =  58  +  98  -  3  a:  +  3  x, 
or  uniting  like  terms,  13  x=  156.  Dividing  both  members  by  13, 
X-  12. 

Proof.     Substitute  12  for  x  in  the  given  equation. 

Then,  2(12-«)  -  3  (9  -  12^  +  5  (12  -  11)  =  7  -  3  (12  -  17), 
or,  8  +  9  +  5  =  7  +  15, 

or,  22  =  22,  an  identity. 

Therefore,  the  value  of  x  is  12. 

Example  3.  Solve  the  equation  14  —  x  —  5  (x  —  3)  (x  -f  2) 
+  (5  -  x)  (4  -  5  x)  =  45  X  -  76. 

Process.     Simplify,  64  -  25  x  =  45  x  -    76. 

Subtract  45  x,  64  -  70  x  =         -    76. 

Subtract  64,  -  70  x  =         -  140. 

Divide  by  —  70,  x  =  2. 

Notet :  1.  To  verify,  that  is,  to  jncne  the  truth  of  the  result^  substitute  the 
supposed  value  of  the  unknown  number  in  the  given  equation  and  thus  find  if  it 
satisfies  its  conditions. 

2.  In  simplifying  an  equation  the  student  should  be  careful  to  notice  that 
when  the  sign  —  precedes  a  term,  in  removing  the  symbol  of  rggregation,  the 
sign  of  each  term  must  be  changed. 

Exercise  39. 
Solve  the  following  equation.s : 
1.    6a;+  1  =  5a:-f  10;  11  -  7z  =  18a:- 14. 

3.  2r+3=  ir,-(2ar-3);   3  (a:- 2)  +  4  =  4  (3  -  a:). 

4.  7(a:- 18)  =  3(x- 14);  7a:+6-3a:  =  56  + 2.r. 

5.  15  (x  -  1)  +  4  (x  +  3)  =  2{x+  7). 

6.  5  -  3  (4  -  a:)  +  4  (3  -  2  x)  =  0. 


106  ELEMENTS   OF  ALGEBRA. 

48.  If  we  add  the  same  number  to  each  member  of  an 
equation,  or  subtract  it  from  each  member,  the  results  are 
equal,  each  to  each.     Thus, 

Consider  the  equation  x  —  b  =  a.     Adding  b  to  each  side,  we  get, 

X  =  a  -{-  b. 
Consider  the  equation  x  +  b  =  a.     Subtracting  6  from  each  side, 
we  have  x  =  a  —  b. 

In  each  case  b  is  transposed  from  one  side  to  the  other, 
but  its  sign  is  changed.     Hence, 

Any  term  may  he  transposed  from  one  side  of  an  equation 
to  the  other,  provided  its  sign  be  changed. 

Example.  Solve  (a:  +  1)  (x  +  2)  (x  +  Q)  -  (x  -  2)  (x -\-  2) 
=  x^  +  9x^  +  4(7x-l)-{-  (2-x)(3  +  x). 

Process.    Simphfy,      x^ +  8x^ -}-20x-\-\6  =  x» +  8x^-}-27x +  2. 
Transpose,  x^  -  x^  +  8x^  ~  8x^  +  20 x-27 x  =  2  -  16. 
Unite  Hke  terms,  —  7  x  =  —  14. 

Divide  by  —  7,  x  =  2. 

*  Hence,  in  general, 

To  Solve  a  Simple  Equation  of  one  Unknown  Number.     If 

necessary,  simplify  the  equation.  Transpose  all  the  terms 
containing  the  unknown  number  to  one  side,  and  all  other 
terms  to  the  other  side.  Unite  like  terms,  and.  divide  both 
sides  by  the  coefficient  of  the  unknown  number. 

Exercise  40. 

Solve  the  following  equations  : 

1.  12  X  -  20  a:  +  13  -  9  2^  -  259  ;  336  +  (3  :c  -  1 1) 

=  2  (5  2:  -  5)  +  8  (97  -  7  a:). 

2.  62;  +  4a;  =  3:r  +  84;  6a:  +  2(13-2^)  =  3  (17 -a;). 


SIMPLE  EQUATIONS.  107 

3.  2  (a;  +2)  +  182:  =  3(5  +  2;)  +  0;  30  a;  +  20 2:- 15  a; 
+  12  a;  =  2820. 

4  9(2;  _  1)  + 2  (a: -2)  =10  (2 -a:);  2  (a:  +  2) (a: - 4) 
=  a;(2a;+  1)-2L 

5.  6y-2(9-4?/)  +  3(52/-7)  =  10  2/-(4  +  16y+35), 
and  verify. 

6.  2y-(4y-l)  =  5y-(//+l);  56+ 21  a;- 8 (2 a;- 1) 
=  62. 

7.  10  [224  -{x-V  192)]  =  7  (28  +  3a:);  9  (7  +  ^y) 
-4[9-(2-.y)]  =  252y. 

8  25  a:-  V.)  -  [3-{4a:-3}]  =  a:  — (a:  — 5),  and  verify. 

9  20(2~a;)  +  3(a:-7)-2[a;+9-3{9-4(2-a:)}]  =  1. 

10.  (y-2)(7-7/)  +  (7/-5)(y+3)-2(y^l)  +  12  =  0. 

11.  4  (v/  +  5)2  -  (2  y  +  1)2  =  3  (y  -  5)  +  180  ;  2.25  x 
-  1.25  =  3  a:  +  3.75. 

12.  .15?/ +  1.575 -.875?/ =  .0625 2^. 
Query.     In  transposing,  why  change  the  signs  ? 

49.  Known  Numbers  are  represented  by  the  first  letters 
of  the  alphabet,  and  by  figures ;  as,  a,  b,  2  c,  6. 

Unknown  Numbers  are  usually  represented  by  the  last 
letters  of  tlie  alpliabet;  as,  x,  y,  z. 

An  Equation  i.s  a  statement  that  two  expressions  repre- 
sent the  same  number. 

An  Identical  Equation,  or  an  Identity,  i.s  one  which  is 
true  for  all  values  of  the  letters  which  enter  into  it ;  as, 
{a  +  z)  (a  —  a:)  =  a2  —  3^. 


108  ELEMENTS  OF  ALGEBRA. 

The  Roots  of  an  equation  are  the  values  of  the  unknown 
numbers. 

The  Degree  of  an  equation  is  the  power  of  the  unknown 
number,  and  is  determined  by  the  greatest  number  of  un- 
known factors  in  any  term. 

Thus,  X  —  y  =  6  is  an  equation  of  the  Jirst  degree ;  4x^  +  5y  =  3 
and  5  xy  +  2  —  3  X  ave  equations  of  the  second  degree. 

A  Simple  Equation  is  an  equation  of  the  first  degree. 

Miscellaneous  Exercise  41. 

Solve  the  following  equations : 

1.  5(7  +  3:?/)-(23/-3)(l-2?/)-(2,y-3)2  +  (5  +  2/)  -  0. 

2.  {22j+lf  +  {27/-lf=UyQf-4:)+57,  and  verify. 

3.  1.5(26i/-51)-12{l-3y)=7Sy-2[5y-2.6{l-.Sy)]. 

4.  .6a:-.7:r  +  .752:-.875:r  +  15  =  0;  .6y-{.lSy-M) 
=  .2  2/  +  4.45. 

5.  30  ^  -  3  [30  z-{2z-  5)]  =  5{2z-  57)  -  50. 

6.  10  (^  +  10)  -  18  (3  2  -  4)  +  5  (3^-2)  (2  z  -  3) 
=  30  2;^  —  16,  and  verify. 

7.  4.Sy-2{.72y-M)  =  1.6?/ +  8.9;  .5x-.3x-.25 
=  .25a:-l. 

8.  .2  7/-  .16?/  =  .6  -.3;  .5y  -  .2y  =  .3y-  15. 

9.  5.6  y  f  .25  y+  .3y  =  y-'S;  .6  y  -\-  .25  -  A  y  =  l.S 
-.75?/ -.3. 

10.  3)x-  .25  (x-2)~-  .3  (3x-{-  12)?  =  41. 


PROBLEMS  LEADING  TO  SIMPLE  EQUATIONS.      109 


CHAPTER  X. 
PROBLEMS  LEADING   TO  SIMPLE  EQUATIONS. 

50.  The  beginner  will  find  the  model  solutions  of  great  benefit 
ill  forming  statements,  and  he  should  give  them  careful  consideration 
before  attempting  to  solve  any  of  the  problems  in  each  set. 

Exercise  42. 

1.  A  father  is  35  and  his  sod  8  years  old.  In  how 
many  years  will  the  father  be  just  twice  as  old  as  the 
son  ? 

Solution.     Let  x  —  the  number  of  years  required. 
Then  x  +  35  =  the  number  of  years  in  the  father's  age  x  years 

from  now, 
and  X  -f  8  =  the  number  of  years  in  the  son's  age  x  years 

from  now. 
By  the  conditions  of  the  problem,  at  the  expiration  of  x  years 
twice  the  son's  age,  or  2  (x  +  8),  equals  the  father's  age,  or  a:  -f  35. 
Hence,  the   equation    2  (x  +  8)  =  x  +  35,  or  2  x  +  16  =  x  -I-  35. 
Transposing  and  uniting  like  terfns,  x  =  19. 

2.  One  number  exceeds  another  by  5,  and  their  sum  is 
29.     Find  the  numbers. 

3.  The  difference  of  two  numbers  is  14,  and  their  sum 
is  48.     Find  the  numbers. 

4.  A  father  gave  S200  to  his  five  sons,  which  they  are 
to  divide  according  to  their  ages,  so  that  each  elder  son 
shall  receive  $10  more  than  his  next  younger  brother. 
Find  the  share  of  each. 


110  ELEMENTS  OF  ALGEBRA. 

5.    A  father  is  four  times  as  old  as  his  son ;  in  24  years 
he  will  only  be  twice  as  old.     Find  their  ages. 


6.  Divide  50  into  two  parts,  so  that  three  times  the 
greater  may  exceed  100  by  as  much  as  8  times  the  less 
falls  short  of  120. 

Solution.     Let  x  =  the  greater  part. 

Then  50  —  x  =  the  less  part, 

and  3  X  =  three  times  the  greater  part ; 

also,  8  (50  -  x)  =  eight  times  the  less  part. 

But,       3  X  —  100  =  the  excess  of  three  times  the  greater  part  over 
100; 
also,  120—8  (50  — x)  =  the  number  that  eight  times  the  less  lacks  of 
120. 

By  the  conditions,  3  a:  -  100  =  120  -  8  (50  -  x). 

Therefore,  .  x  =  36,  for  the  greater  part, 

and  50  —  a;  =  14,  for  the  less  part. 

7.  Twenty-three  times  a  certain  number  is  as  much 
above  14  as  16  is  above  seven  times  the  number.  Find 
the  number. 

8.  A  is  five  years  older  than  B.  In  15  years  the  sum 
of  their  ages  will  be  three  times  the  present  age  of  A. 
Find  the  age  of  each. 

9.  A  is  25  years  older  than  B,  and  A's  age  is  as  much 
above  20  as  B's  is  below  85.     Find  their  ages. 

10.  The  sum  of  the  ages  of  A  and  B  is  30  years,  and 
five  years  hence  A  will  be  three  times  as  old  as  B.  Find 
their  ages. 

11.  The  difference  between  the  squares  of  two  consecu- 
tive numbers  is  121.     Find  the  numbers. 


PROBLExMS  LEADUSG  TO  SIMPLE  EQUATIONS.       Ill 

Solution.     Let       x  =  the  less  number. 
Then  will  jc  -f  1  =  the  greater  number, 

x^  =  the  stjuare  of  the  less  number, 
and  (x  -I-  1)*^  =  the  stpuire  of  the  greater  number. 

Then     (x  +  1)^  -  x'^  =  the  ditlerence  of  the  scjuai-e  numbers. 
But  121  =  the  difference  of  the  squares. 

Hence,  (x  +  l)^  -  x-  =  121. 
Therefore,  x  =  60,  the  less  number, 

X  -h  1  =  61,  the  greater  number. 

12.  Find  three  consecutive  numbers  whose  sum  is  27. 

13.  The  difference  of  two  numbers  is  3,  and  the  differ- 
ence of  their  squares  is  21.     Find  the  numbers. 

14.  Find  a  number  such  that  if  5,  15,  and  35  be  added 
to  it,  the  product  of  the  first  and  third  results  may  be  equal 
to  the  square  of  the  second. 

15.  I  sold  a  cow  for  S35  and  half  as  much  as  I  gave  for 
it,  and  gained  SIO.     Find  the  cost  of  the  cow. 


16.  A  had  four  times  as  much  money  as  B ;  but,  after, 
giving  B  $16,  he  had  only  two  times  as  much  as  B.  How 
much  had  each  at  first  ? 

Solution.     Let  x  =  the  number  of  dollars  that  B  had  at  first. 
Then  4x  =  the  number  of  dollars  that  A  had  at  first. 

But  4x  —  16  =  the  number  of  dollars  that  A  had  after  giving 

B^16, 
and  X  +  16  =  the  number  of  dollars  B  had  after  receiving 

^16  from  A. 
By  the  conditions,  4  X- 16  =  2  (x  +  16). 

Therefore,  x  =  24,  the  number  of  dollars  that  B  had, 

and  4x  =  96,  the  number  of  dollars  that  A  had. 

17.  A  father  is  3  times  as  old  as  his  son ;  four  years 
ago  the  father  was  4  times  as  old  as  bis  son  then  was. 
Find  their  ages. 


112  elp:ments  of  algebra. 

18.  One  number  is  two  times  another;  but  if  50  be 
subtracted  from  each,  one  will  be  three  times  the  other. 
Find  the  numbers. 

19.  A  has  $26.20  and  B  has  $35.80.  B  gave  A  a  cer- 
tain sum;  then  A  had  four  times  as  much  as  B.  How 
much  did  A  receive  from  B  ? 

20.  If  288  be  added  to  a  certain  number,  the  result  will 
be  equal  to  three  times  the  excess  of  the  number  over  12. 
Find  the  number. 

21.  A  farmer  has  grain  worth  $0.60  per  bushel,  and 
other  grain  worth  $1.10  per  bushel.  How  many  bushels 
of  each  kind  must  be  taken  to  make  a  mixture  of  40 
bushels  worth  $0.90  a  bushel? 

Solution. 

Let  X  =  the  number  of  bushels  required  of  the  f  0.60  grain. 

Then  40  —  a:  =  the  number  of  bushels  required  of  the  $1.10  grain; 

and  Y®^*\j  X  —  the  number  of  dollars  in  the  cost  of  the  $0.60  grain; 

also,  1.10(40-a:)  —  the  number  of  dollars  in  the  cost  of  the  $1.10grain. 

Hence,  ^-^^  a;  -f-  1.10  (40  —  x)  —  the  number  of  dollars  in  the  total 

cost  of  the  mixture. 
But  the  cost  of  the  mixture  is  to  be  $36.     Hence, 

^%x-\-  1.10  (40 -a:)  =  36. 
Therefore,      x  =  16,  the  number  of  bushels  of  the  $0.60  kind, 
and  40  -  X  =  24,  the  number  of  bushels  of  the  $1.10  kind. 

22.  A  merchant  has  two  kinds  of  vinegar :  one  worth 
$0.35  a  quart  and  the  other  $1.25  a  gallon.  From  these 
he  made  a  mixture  of  63  gallons,  worth  $1.30  a  gallon. 
How  many  gallons  did  he  take  of  each  kind  ? 

23.  A  merchant  has  a  mixture  of  88  pounds  of  13  and 
11  cent  sugar,  which  he  sells  at  12|  cents  per  pound. 
How  many  pounds  of  each  kind  are  there  ? 


PROBLEMS  LEADING  TO  SIMPLE  EQUATIONS.      113 

24.  I  bought  24  pounds  of  tea  of  two  different  kinds, 
and  paid  for  the  whole  $9.  The  better  kind  cost  $0.65 
per  pound,  and  the  poorer  kind  $0.35  per  pound.  How 
many  pounds  were  there  of  each  kind  ? 

25.  A  grocer  having  75  pounds  of  tea  worth  $0.90  a 
pound,  mixed  with  it  so  much  tea  at  $0.50  a  pound  that 
the  combined  mixture  was  worth  $0.80  a  pound.  How 
much  did  he  add  ? 

Remarks.  No  general  method  can  be  given  for  the  solution  of 
problems.  ^ 

The  beginner  will  find  that  his  principal  difficulty  in  solving  a 
problem  consists  in  forming  the  equation  of  conditions,  and  in  order 
to  overcome  this,  much  will  depend  upon  his  skill  and  ingenuity. 

The  statement  of  a  problem  consists  in  translating  its  conditions 
into  algebraic  symbols  and  ordinary  language.  Many  times  the  be- 
ginner fails  to  form  a  correct  statement,  because  he  does  not  under- 
stand what  is  meant  by  the  ordinary  language  of  the  problem.  If  he 
cannot  assign  a  consistent  meaning  to  the  words,  it  will  be  impossi- 
ble for  him  to  express  their  meaning  in  algebraic  symbols.  It  often 
happens  that  the  words  appear  to  be  susceptible  of  more  than  one 
meaning.  In  such  caj^es  the  student  should  express  the  meaning  that 
seems  most  reasonable  in  algebraic  symbols,  and  obtain  the  result  to 
which  it  will  lead.  Should  such  result  be  inadmissible,  the  student 
should  Uy  another  meaning  of  the  words. 

The  student  must  depend  upon  hus  own  powers^  and  should  he  at 
times  be  perplexed,  he  must  not  be  discouraged,  since  nothing  but 
patience  and  practice  can  overcome  the  difficulties  and  give  him 
readiness  and  certainty  in  solving  problems.  He  must  study  the 
iT»eaning  of  the  language  of  the  problem,  to  ascertain  the  unknown 
numbers  in  it.  There  may  be  several  such  numl^rs,  but  oftentimes  a 
little  skilful  manipulation  will  enable  one  to  express  all  of  the  un- 
known numbers  in  terms  of  some  one  of  them.  Select  the  one  by 
which  this  can  be  most  easily  done  and  represent  it  by  some  one  of 
the  final  letters  of  the  alphabet. 

Among  the  following  problems  no  doubt  the  beginner  will  find 

8 


114  ELEMENTS  OF   ALGEBRA. 

some  which  he  can  readily  solve  by  arithmetic,  or  by  guessing  and 
trial;  he  may  thus  be  led  to  undervalue  the  power  of  algebra,  and  to 
regard  its  aid  as  unnecessary.  In  reply,  as  the  student  advances  lie 
will  find  that  by  the  aid  of  algebi'a  he  can  solve  not  only  all  of  these 
problems,  without  any  uncertainty  or  guessing,  but  those  which 
would  be  exceedingly  difficult,  if  not  altogether  impossible,  if  he 
depended  upon  arithmetical  processes  alone. 

26.  A's  age  is  six  times  B's,  and  fifteen  years  hence  A 
will  be  three  times  as  old  as  B.     Find  their  ages. 

27.  A  is  three  times  as  old  as  B,  and  12  years  since  he 
was  fiv^  times  as  old.     Find  B's  age. 

28.  A  father  has  three  sons;  his  age  is  60,  and  the 
joint  ages  of  the  sons  is  46.  How  long  will  it  be  before 
the  joint  ages  of  the  sons  will  be  equal  to  that  of  the 
fathei'  ? 

29.  If  yon  walk  10  miles,  then  travel  a  certain  distance 
by  train,  and  then  twice  as  far  by  coach,  and  the  whole 
journey  is  70  miles,  how  far  will  you  travel  by  coach? 

30.  A  is  twice  as  old  as  B,  and  seven  years  ago  their 
united  ages  amounted  to  as  many  years  as  now  represent 
the  age  of  A.     Find  their  ages. 

31.  After  136  quarts  had  been  drawn  out  of  one  of  two 
equal  casks,  and  80  gallons  out  of  the  other,  there  remained 
just  three  times  as  much  in  one  cask  as  in  the  other. 
Find  the  contents  of  each  cask. 

32.  Find  the  number  whose  double  increased  by  1.2 
exceeds  3.65  by  as  much  as  the  number  itself  is  less  than 
8.65. 

33.  Find  three  consecutive  numbers  such  that  if  they 
be  diminished  by  10,  17,  and  26,  respectively,  their  sum 
will  be  10. 


PROBLEMS  LEADING  TO  SIMPLE  EQUATIONS.      115 

34.  Two  consecutive  numbers  are  such  that  one  fourth 
of  the  less  exceeds  one  tifth  of  the  greater  by  1.  Find  the 
numbers. 

35.  There  are  two  consecutive  numbers  such  that  one 
fifth  of  the  greater  exceeds  one  tenth  of  the  less  by  3. 
Find  tliem. 

36.  Find  a  number  such  that  the  sum  of  its  half  and 
its  fourth  shall  exceed  the  sum  of  its  fifth  and  its  tenth  by 
45. 

37.  Find  a  number  such  that  the  sum  of  its  half  and 
its  fifth  shall  exceed  the  difiference  of  its  fourth  and  its 
tenth  by  110. 

38.  If  a  watch  and  chain  are  worth  $185,  and  the 
watch  lacks  $19  of  being  worth  two  times  the  cost  of  the 
chain,  find  the  cost  of  each. 

39.  If  silk  costs  6  times  as  much  as  linen,  and  I  buy 

22  yards  of  silk  and  28  yards  of  linen  at  a  cost  of  $52, 
find  the  cost  of  each  per  yard. 

40.  A  man  gave  17  boys  $3.31,  giving  to  some  13  cents 
each  and  to  the  rest  23  cents  each.     How  many  received 

23  cents  ? 

41.  I  paid  a  bill  of  $1.53  with  39  pieces  of  money, 
some  3-cent  and  the  rest  5-cent  pieces.  How  many  of 
each  did  it  take  ? 

42.  A  son  earns  37  cents  per  day  less  than  his  father, 
and  in  8  days  the  father  earns  $6.08  more  than  the  son 
earns  in  5  days.     Find  the  daily  wages  of  each. 

43.  How  many  10-cent  pieces  and  how  many  25-cent 
pieces  must  be  taken  so  that  95  pieces  shall  make  $12.35? 


116  ELEMENTS  OF  ALGEBRA. 

44.  Divide  $112  into  two  parts,  so  that  the  number  of 
five-cent  pieces  in  one  may  equal  the  number  of  three-cent 
pieces  in  the  other. 

45.  A  sum  of  money  consists  of  dollars,  twenty-five- cent 
pieces,  and  dimes,  and  amounts  to  $29.50.  The  number 
of  coins  is  55.  There  are  twice  as  many  dimes  as  quarters. 
How  many  are  there  of  each  kind  ? 

46.  A  sum  of  £8  17  s.  is  made  up  of  124  coins,  consist- 
ing of  florins  and  shillings.     How  many  are  there  of  each? 

47.  A  bill  of  £4  5s.  was  paid  in  crowns,  half-crowns, 
and  shillings.  The  number  of  half-crowns  used  was  four 
times  the  number  of  crowns  and  twice  the  number  of  shil- 
lings.    How  many  were  there  of  each  ? 

48.  A  bill  of  £48 J  was  paid  with  guineas  and  half- 
crowns,  and  12  more  half-crowns  than  guineas  were  used. 
How  many  were  there  of  each  ? 

49.  A  company  of  84  persons  consists  of  men,  women, 
and  children.  There  are  three  times  as  many  men  as 
women,  and  five  times  as  many  women  as  children.  How 
many  are  tliere  of  each  ? 

50.  The  sum  of  three  numbers  is  263.  The  first  is  3 
times  the  second,  and  the  third  is  23  more  than  5  times 
the  sum  of  the  other  two.     Find  the  numbers. 

51.  A  farmer  wishes  to  mix  660  bushels  of  feed,  con- 
taining oats,  corn,  rye,  and  barley,  so  that  the  mixture 
may  contain  two  times  as  much  corn  as  oats,  three  times 
as  much  rye  as  corn,  and  four  times  as  much  barley  as 
rye.     How  many  bushels  of  each  should  be  used  ? 


PROBLEMS  LEADING  TO  SIMPLE  EQUATIONS.      117 

52.  Divide  $2590  into  two  sucli  parts  that  the  first  at 
7%  simple  interest  for  8  years  may  amount  to  the  same 
sum  as  the  second  in  5  years  at  8  %. 

Note.  The  character  %  is  sometimes  used  for  the  term  "/>er  cent."  Per 
cent  is  used  by  ellipsis  for  rate  per  cent.  Thus,  au  allowauce  of  7  on  a  hundred 
is  at  a  rate  of  .07,  aud  the  rate  per  cent  is  7. 

53.  $330  is  invested  in  two  parts,  on  one  of  which 
15%  is  gained,  and  on  the  other  8  %  is  lost.  The  total 
amount  returned  from  the  investment  is  S345.  Find  the 
investment. 

54.  A  man  has  $  7585.  He  built  a  house,  aud  put  tho 
rest  out  at  simple  interest  for  18  months;  40%  of  it  at 
5  %  and  the  remainder  at  6  %.  The  income  from  both  in- 
vestments is  $211.26.     Find  the  cost  of  the  house. 

55.  In  a  certain  weight  of  gunpowder  the  saltpetre  was 
4  pounds  less  than  half  the  weight,  the  sulphur  5  pounds 
more  than  a  fifth,  and  the  charcoal  3  pounds  more  than 
a  tenth.     Find  the  number  of  pounds  of  each. 

56.  A  company  of  266  persons  consists  of  men,  women, 
aud  children.  Tlie  men  are  14  more  in  number  than  the 
women  ;  the  children  34  more  than  the  men  and  women 
together.     How  many  are  there  of  each  ? 

57.  I  bought  16  yards  of  cloth,  and  if  I  had  bought  one 
yard  less  for  the  same  money,  each  yard  would  have  cost 
$0.25  more.     Find  the  cost  per  yard  of  the  cloth. 

68.  A  and  B,  85  miles  apart,  set  out  at  the  same  time 
to  meet  each  other;  A  travels  5  miles  an  hour  aud  B  4 
miles  an  hour.  How  far  will  each  have  travelled  when 
they  meet  ? 


118  ELEMENTS   OF  ALGEBRA. 

59.  $330  is  loaned  for  nine  months  in  two  parts ;  on 
one  15  %  per  annum  is  gained,  and  on  the  other  8  % 
per  annum  is  lost.  The  total  amount  from  the  loan  is 
$364.25.     Find  the  amount  in  each  loan. 

60.  A  boy  has  a  certain  sum  of  money,  lie  borrowed  as 
much  more,  and  spent  12  cents;  he  again  borrowed  as 
much  as  he  had  left,  and  spent  12  cents ;  again  he  bor- 
rowed* as  much  as  he  had  left,  and  spent  12  cents ;  after 
which  he  had  nothing  left.  How  much  money  had  he  at 
first? 

61.  A  carriage,  horse,  and  harness  are  worth  $720.  The 
carriage  is  worth  eight  tenths  of  the  value  of  the  horse,  and 
the  harness  six  tenths  of  the  difference  between  the  value 
of  the  horse  and  carriage.     Find  the  value  of  each. 

62.  A  boy  sold  half  an  apple  more  than  half  his  apples. 
Again  he  sold  half  an  apple  more  than  half  his  remaining 
apples.  A  third  time  he  repeated  the  process;  and  he  had 
sold  all  his  apples.     How  many  apples  had  he  ? 

Algebra  is  the  science  which  treats  of  algebraic  liumbers 
and  the  symbols  of  relation. 

Algebra,  like  arithmetic,  is  a  science  which  treats  of  numbers.  In 
arithmetic  the  numbers  are  positive  and  represented  by  figures.  In 
algebra  the  letters  of  the  alphabet  or  figures  are  used  to  represent 
numbers,  and  they  may  be  positive  or  negative,  real  or  imaginary. 

Algebra  enables  us  to  prove  general  theorems  respecting  numbers, 
and  also  to  express  those  theorems  briefly. 


FACTORING.  119 

CHAPTER  XL 
FACTORING. 

51.   A  Factor  is  one  of  the  makers  of  a  number. 

Thus,  since  5  with  the  aid  of  4  and  by  the  process  of  multiplica- 
tion makes  20,  5  is  a  factor  of  20. 

A  factor  is  also  a  divisor,  but  it  is  considered  a  divisor  when  it 
separates  a  number  into  parts,  not  when  it  helps  to  make  up  a 
number. 

Note.     Unity  cannot  be  a  factor. 

Factoring  is  the  process  of  separating  an  expression  into 
its  factors. 

Example.     Find  the  factors  of  12  a«  6  x^. 

Solution.  The  prime  factors  of  12  are  2,  2,  and  3.  The  factors 
of  a'  are  a,  a,  and  a.     The  factors  of  x^  are  x  and  a:*. 

Therefore,  l2a*bxi  =  2X2X3xaXaXaXbXxXxK 
Hence,  a»  a  direct  result  of  the  principle  that  monomials  are  mul- 
tiplied by  writing  the  several  letters  in  connection,  and  giving  each 
an  exponent  equal  to  the  sum  of  the  exponents  of  that  letter  in  the 
factors. 

To  Factor  a  MonomiaL  Separate  the  letters  into  any  number 
of  factors,  so  that  the  sum  of  all  the  exponents  of  each  factor  shall 
make  the  exponent  of  that  factor  in  the  given  expression  ;  also  sepa- 
mte  the  numerical  coefficient  into  its  prime  factors. 

Exercise  43. 

Separate  into  factors  with  integral  exponents : 

1.    Ua^l^x;   Wo^t/^;    15ah^(^;  20ab(^;  Soa^f:!^^; 


120  ELEMENTS  OF  ALGEBRA. 

Separate  into  two  equal  factors : 

2.  UaH'^;  da^f;  SI  a^  b^  x^""  y^"" ;  169a"5. 
Eemove  the  factor  2  a^  bi  from  : 

3.  Sa^b;  Qabx;  IQab^c^;  10 a'H-^ x^y^. 
Separate  into  three  factors,  also  into  four : 

4.  cc ;  m^ "  ;  a"  ;  xi  ;  x^. 

52.    Example  1.     Factor  a^x  -  Sa^x^. 

Solution.  Dividing  the  expression  by  a^  x,  we  have  a  —  3  aj. 
Hence,  a^  x  —  3a^x  =  a'^x  {a  —  3  x). 

Example  2.     Factor  5 a'^b^x^  -  15 ab^x^  +  20 b« x^. 

Solution.  By  examining  the  terms  of  the  expression  we  find 
that  5  b^  x^  is  a  factor  of  every  term.  Dividing  by  this  common  fac- 
tor the  other  is  fomid.  Hence,  the  factors  are  5  h^x^  and  a^x  —  Sax 
+  4  6. 

.-.  5aH^x^-l5ab^x»-\-20b^x^  =  5b^x^(a^x-3ax  +  4b). 
Hence, 

When  the  Terms  of  a  Polynomial  have  a  Monomial  Factor. 

Divide  each  term  of  the  expression  by  the  common  factor.     The 
divisor  and  quotient  will  be  the  required  factors. 

Exercise  44. 

Factor  the  following : 

1.  7n^  +  n;  4:  a^b  +  ab^c+ Sab;  Sa^- 12  a^. 

2.  ax  —  bx+cx;S9a^7/-{-57x^y^. 

3.  5x^  +  Sa^-x^;  72  h^  x'^y^  -  84:b^  x  y^  -  9Q  a  b  a^y^. 

4.  924  «2  x^y^'z-  1 178  a  x""  y  z""  +  1232  a^  x""  y'^  z\ 


FACTORING.  121 

5.  4:aH-(J0aI^+20abc-{-SaH*x^+Uahy-Z6aHcx^, 

6.  2  xi  y  ^  a  b  X  y  +  c  a^  y^ ;  5  x^  +  10  x^  —  15  xi. 

53.  In  certain  Trinomiala,  of  the  form  x^  -\-  ax  -^  b,  where  a 
and  b  represent  any  numbers,  either  integral,  fractional,  positive,  or 
negative,  it  is  possible  to  reverse  the  operation  of  Art.  25,  and  sepa- 
rate the  expression  into  the  product  of  two  binomial  factors.  Evi- 
dently the  first  term  of  each  factor  will  be  the  square  root  of  a;*,  or  x; 
and  to  obtain  the  second  terms  of  the  factors,  /ind  two  numbers  whose 
algebraic  product  is  the  last  term,  or  b,  and  whose  sum  is  the  coefficient 
ofx,  or  a. 

Example  1.    Factor  x^  +  21  a:  -{-  110. 

Solution.  Evidently  the  first  term  of  each  factor  will  be  x.  The 
second  term  of  the  factors  must  be  two  numlnirs  whose  product  is 
110  (the  third  term),  and  whose  sum  in  21  (the  coefficient  of  a;).  The 
only  two  numbers  whose  product  is  110  and  whose  sum  is  21  are  10 
and  11.     Therefore,  z«  +  21 1+  1 10  =  (a:  +  10)  (x  +  U). 

Example  2.     Factor  x^  +  x-  132. 

Solution.  Evidently  the  first  term  of  each  binomial  factor  will 
be  X.  The  second  term  of  the  two  binomiul  factors  must  be  two 
numbers  whose  algebraic  product  is  —  132  and  whose  sum  is  -f-  1 
(the  coefficient  of  x).  The  only  two  numbers  whose  product  is  — 132 
and  whose  sum  is  -f  1  are  4- 12  and  —  11.  Therefore,  x^ -\-  x  —  132 
=  (X -\- U)  (x  -  U). 

Example  3.     Factor  y^  -  5cy  —  50 c^ 

Solution.  Evidently  the  first  terra  of  each  binomial  factor  will 
be  y.  The  second  term  of  the  two  binomial  factors  must  Ijc  two 
numbers  whose  product  is  —  50  c*  and  whose  sum  is  —5c  (the  coef- 
ficient of  y).  The  only  two  numbers  whose  pro<luct  is  —  50  c*  and 
whose  sum  is  —5c  are  +  5  c  and  —  10  c.  .-.  y^  —  5  cy  —  50  c* 
=  (y-l-5c)(y-10c). 


122  ELEMENTS   OF   ALGEBRA. 

Example  4.     Factor  x^y^  —  (jn  —  n)  xij  —  m  n. 

Solution.  Evidently  the  first  term  of  each  binomial  factor  will 
ho,  xy..  The  second  term  of  the  two  binomial  factors  must  be  two 
such  numbers  whose  product  is  —nin  and  whose  sum  is  —  (in  —  n). 
The  only  two  numbers  whose  product  is  —  m  n  and  whose  sum  is 
—  (m  —  n)  are  +  n  and  —  m. 

.*.  x^y'^  —  (in  —  n)  xy  —  m  n  =  (x  y  -\-  n)  {xy  —  m).     Hence, 

I.   If  the  Coefficient  of  the  Highest  Power  is  Unity.   For  the 

first  term  of  each  factor  take  the  square  root  of  one  term  of  the  trino- 
mial ;  and  for  the  second  term  of  the  factors,  such  numbers  that 
their  algebraic  product  will  be  another  term  of  the  trinomial,  and 
their  sum  multiplied  by  the  first  term  of  either  factor  will  be  the 
remaining  term  of  the  trinomial. 

Exercise  45. 

Factor  the  following  : 

1.  ^24.19^+88;    .^2-72^+  12;    ft8_20a4  +  96. 

2.  2:2 +  35  a; +  216;  52(^2- 245c +  143. 

3.  ft* ¥  +  37 a2  62  +  300  ;  a^  +  5  ah  -  66  h\ 

4.  a?ly^-oah-24.-  ft*  +  15*2  +  44.  a^  +  17  ft^  +  60. 

5.  a^y---dahc-lQc^;  a^-2a'^-120;  yi2+.8  ^i  +  l.o. 

7.  ^2  _  15  a;  +  44;  ^„2  +  .i_i.  ,„,  ^  i^_ .  ^2  _  11  ^^  _  26. 

8.  130  +  31  ft  &  +  ft2  /;2  .  ,,2  -  20  ah  x+  75  h'^  oj^  ;  y^ 
+  6  a;2  7/2  _  27  2)4;  1  +  13  ^  +  42  :r2  ;  ^;i2  -  15  a  m  +  06  a^. 

9.  ft2  -  lSax7j  -  243  x^  y^;  (x  +  yf  +  5  {x  +.  ?/)  +  4. 

10.  40  ft2  2,2  _  13  ,-^  7,  +  1  .  (^  _  5)2  +  (^  -h)-2. 


FACTORING.  123 

11.  (x-yf-d{x-y)-l0;ay^+54x-]'729',  204-29^^+ .,4. 

12.  (a  -hiy  -^9  (a  +  i)2  +  8  ;  2;*"  -  (6  +'m)  x"^""  -^  b  vi. 

13.  a2  -  10  a  V^v  -  39  h^c^-  x^- -^  {a  -  h)  x""  -  a  b. 

14.  a^-9xij-70f',  a;2-.|.c-|;  2;*''-43ic2n^46Q 

15.  2^-{-iax--^^a^;  x*-a^x'^-4e2a\ 

16.  x^i/-^Sx7/-154:;  a^"  x*"'  +  14  a"  x^"'  2/"  +  33  y^. 

17.  a.>2  ?/2  _  28  a"  i"  a;?/  +  187  a^-  b^-  -  x^^\x  +  '^^V- 

18.  a;*"//"  +  20a'"^/'";;t2''7/2n  ^  5irt2«.j2m.  (^  ^  ^^^em 
--  7  a*"  (a;  +  ?/)3-  -  98  a^"  ;  n*  +  .01  n^  -  .011. 

19.  ^+^x-^\;  u;2+2^2/~.21y2;  «4+_8^^2+ j^. 

By  an  extension  of  the  foregoing  principles  we  may  factor  some 
trinoniials,  of  the  form  c^x^  +  ax  -\- hd^  where  the  coefficient  of  a:^  is 
a  perfect  square.     Thus, 

Example  20.     Factor  4  a:«  +  4  a:  -^  3. 

Solution.  The  first  term  of  each  binomial  factor  will  be  the 
square  root  of  4  x^.  The  second  term  of  the  two  binomial  factors 
must  >>e  two  numbers  whose  product  is  —  .3  and  whose  sum  multiplied 
by  2  a;  is  +  4  x.  The  only  two  numbers  whose  product  is  —  3  and 
whose  sum  multiplied  by  2  a:  is  +  4  a;  are  +  3  and  —  1 . 

.-.  4a:«  +  4a;-3=(2x  +  .3)(2a:-l). 

21.  4a.^-10:r  +  6;  9a^»-27a;+18;  4a'2+16aa;+12a2 

22.  9  rt2  +  30  «  5  +  24  J2;  16  a^«  -  20  a  a;  +  6  «« 

23.  25.xiO'"-|2r5"'rt''-Ja2-;  36(a-t)*-  +  12(a-6)*"  +  * 
-  143  (a  -  h)\ 


124  ELEMENTS  OP  ALGEBRA. 

54.    We  may  factor  some  trinomials  of  the  form  ax^  +  bx  ■{-  c. 

Thus, 

Example  1.     Factor  8  a;^  -  38  a;+  35. 

Solution.  The  first  term,  8x%  is  the  product  of  the  first  terms  of 
the  binomial  factors.  The  last  term,  35,  is  the  product  of  the  second 
term  of  the  two  binomial  factors.  It  is  evident  that  the  first  term  of 
each  binomial  factor  might  be  ±2 a:  and  ±.4x,  or  ±Sx  and  ±a:; 
also  the  last  terms  of  the  two  factors  might  be  ±  7  and  ±  5,  or  db  35 
and  ±  1.  From  these  w-e  must  select  those  that  will  produce  the 
middle  term,  —38  x,  of  the  trinomial.  Since  (+2x)  X  (—  5)  +  (+  4 cc) 
X  (—  V)  =  —  38  X,  we  must  take  +  2  a:  and  +  4  x  for  the  first  terms, 
and  —  7  and  —  5  for  the  corresponding  second  terms  of  the  two  bino- 
mial factors.     Therefore,  8  a;^  -  38  a;  +  35  =  (2  a;  -  7)  (4  ic  —  5). 

Example  2.     Factor  6x*  ~  bx'^y^  —  6  y\ 

Solution.  Take  +  3  a;^  and  +  2  x^  for  the  factors  of  6  x*,  and 
+  2  y'^  and  —  3  2/^  as  those  of  —  6  ?/*.  We  now  arrange  them  in  bino- 
mial factors,  so  that  the  algebraic  sum  of  their  cross  products  shall  be 
-  5  a:2 1/2.  Since  (+  3  a^)  X  (-  3  7/2)  +  (+  2  x^)  X  (+2y^)  =  -5x^ y\ 
+  3  a;2  and  +  2  a;^  are  the  first  terms,  and  +  2  y'^  and  —3y^  are  the 
corresponding  second  terms  of  the  factors.  .•.  6x^  —  6  x^y^  —  6y* 
=  (3  a:2  +  2  ?/)  (2  a:2  -  3  y^).     Hence, 

II.   If  the  Coefficient  of  the  Highest  Power  is  not  Unity 

Arrange  the  trinomial  in  descending  powers  of  a  common  letter. 
Select  factors  of  the  extreme  terms  and  arrange  them  in  binomial 
factors,  so  that  the  algebraic  sum  of  their  cross  products  shall  be  the 
second  term  of  the  trinomial. 


Exercise  46. 

Factor  the  following : 

1.  4  x^  +  l.S  a:  +  3;  4  2/2  _  4  y  _  3;  12  a^ -\-  a^a^  -  x\ 

2.  S  +  llx-4:X^;  Sx^'-22xy-21f;  ^o?'x^  +  ax-^l. 


FACTORING.  125 

3.  8  m6  -  19  m3  _  27;  15  a^  _  58  a  +  11  ;  6  a2  ^.  ^ ah 
-  3  62;  2  //<2  _  13  m  71  +  6  n2  ;  3  a^»  +  7  a;  +  4. 

4    24  +  37a-72a2;  15^:24.  224a: -15;  4-5a;-6a:2. 

5.  6a^»- 19.ry  +  10/;  %  a^ -^^  U  xy  -  Ibf)  lb  a^ 
-77  a; +10;  24  a,^  +  22  2;- 21  ;  lla2  +  34a4-  3 

6.  18-33a;+5ar';  Ga:2_7a,2^_3^.  5  +  32;«- 21a;2. 

7.  24a,'2-29a;y-4y2;  ea^^+lQ^ra-yn^y^^m 

8.  2(a;+y)2+5(a;+y)(m  +  ?i)4-2(m  +  n)a;  2«2+a;-28. 

9.  2(x+yf-l{x^y){a^h)  +  Z{a  +  hf-  l^x'j^^x-^. 

10.  n{x-yf''-l'6x'^y^{x-ijf''^2x^'^f)  27a2+6a-l. 

11.  8  a2"  -f  34  rt»  (2:  -  yY^  +  21  (a:  -  ;/)2'»». 

55.  A  trinomial  is  a  perfect  square  when  two  of  its  terms  are 
positive,  and  the  third  term  is  twice  the  product  of  their  .square  roots. 
Such  trinomials  are  particular  forms  of  I.,  and  their  binomial  factors 
are  equal. 

Example.     Factor  4  z^  +  44  a:y  +  121  y\ 

Solution.  The  first  term  of  each  binomial  factor  will  be  the 
8C[uare  root  of  4  x^,  or  2  a:  and  2x,  For  the  second  terms  of  the  bino- 
mial factors  tjike  the  square  root  of  121  ?/*,  or  11  ?/  and  11  y.  Since 
the  terms  of  the  trinomial  are  positive  the  factors  are  2  x  +  1 1  y  and 
2x  +  lly.     Therefore, 

4x«  +  44xy4-  121  y2=  (2x+lly)(2x+  11  y) 
=  (2x4-lly)^.     Hence, 

III.  If  the  Trinomial  is  a  Perfect  Square.  Arrange  the  tri- 
nomial .according  to  the  powers  of  one  letter.  For  one  of  the  equal 
factors,  fin<l  the  square  roots  of  the  first  and  last  terms,  and  connect 
these  roots  by  the  sign  of  the  second  term. 


126  ELEMENTS  OF  ALGEBRA. 

Exercise  47. 

Factor  the  following : 

+  225&2c2;   a6-4a4  +  4a2 

2.  49 m6- 140  m%2 +100714;  si  x"^ 9/ -126  a^xhj  + 49 a^. 

3.  7n}^—2m^ni-n'^;  l-10mn  +  25m^n^;  x'^+2x^i-x^. 

4.  {a  +  hf  +  16  (a  +  ^>)  +  64 ;  7?i2  +  18  m  +  81. 

5.  4a*a^-20a2^7/  +  25a;42/^  o61aH^c^-16ahcdmn 
+  4:d^m^'n?;  121  7/2,27^4  -  220  mn'^2^  +  100^2 

6.  225  X^  -  30  a;2  7/2  +   ^4  .    4  ^4«  _  4  ^2n  ^m   _^   ^2m^ 

7.  49  m27i2  +  -2^  m  n^  +  ^71*;  ^j2  +  ^  +  1. 

8.  -^^a^  +  ^^^W  +  \a^h^-  a^c  +  6a^h^e+9Wc. 

9.  9^:2  _  3:^7/  +  ^7/2;   (m  -  7O2  +  2  (7?z,  -  ^i)  +  1. 

10.  {ci?-af-\.6{a^-a)+  9;  4 (a;  +  7/)2  +  Jg- +  a?  +  7/. 

11.  at  +  6l  —  2  aHf ;  7?i  —  2  ??ii  +  1 ;  ??z2  71  +  77^  7i2  —  2  77it  77!. 

12.  x-\-2  x^y^  +  7/ ;  7?i2  n  +  a2—  2  a  m  n^ ;  4  ic+ 1 2  71  a:^  +  9  7i2. 

13.  (a  +  2>)2"-10(a  +  &)"c  +  25c2;  |  ^5m_^  _i_6___  11  J.«. 

56.    Example.     Factor  8  a;^  -  27  i/^. 

Solution.  Evidently  (Art.  34)  2  re  -  3  y  is  a  divisor  of  8  x^  -  27 1/^. 
Dividing  8a:^  l)y2x,  we  have  4x^,  the  first  term  of  the  quotient. 
Divide  4  rc^  by  2  oj,  multiply  the  result  by  2y,  and  we  have  Qxy,  the 
second  term  in  the  quotient.  In  like  manner  we  find  9  .y^  for  the  last 
term  in  the  quotient.  Hence,  the  quotient  is  ^x^  +  Qxy  +  ^  y^. 
Therefore,  the  factors  of  the  binomial  are  2ic  — 3?/  and  4x'^+  6a:?/  +  9?/2. 


FACTORING.  127 

Since  the  dividend  is  equal  to  the  prcKluct  of  the  divisor  and  quotient, 
:i*  —  27 y*  =  (2 X  —  3 y)  {4  x^  -h  6 xy  +  d y^).     Hence,  in  general, 

When  a  Binomial  is  the  Difference  of  Two  Equal  Odd 
Powers  of  Two  Numbers.  Cunsider  the  binomiid  a  dividend, 
and  find  a  divisor  and  quotient  by  inspection  (Art.  34)  The  divisor 
and  quotient  will  be  the  required  factors. 

Exercise  48. 

Factor  the  following : 

1.  l-d'i'Sa^;  82^-7297/)  216x^-a^ 

2.  2^,/  -  aH^;  x^  -1;  243  a^  -  b^;  a^h^  -  m^ 

3.  216  d?  —  343  ;  3  a;  —  81  a:*.  Suggestion.  Remove  the 
monomial  factor  3x  first. 

4   a}^  -  1024  6^0 ;  729  x^  -  1728  f\x-^-  y-\ 

5.  135a;*-320ar2;  2an-64a&;  a;-^-7/-f. 

6.  a655-2:6/;  64^6-125^3;  a:3n_^« 

57.    Example.     Factor  729  +  a«. 

Solution.  Since  729  is  the  6th  power  of  3,  3*  +  r|2  (Art.  36)  is 
a  divisor  of  729  +  a'-  Dividing  729  by  3^*  we  have  3*,  the  fii-st  term 
in  the  quotient.  Divide  3*  by  3',  multiply  the  result  by  a'*,  and  we 
have  3* a*,  the  ."^econd  term  in  the  quotient.  In  like  manner  we 
find  a*  for  the  last  term  in  the  quotient.  Honcc,  the  quotient  is 
3*  -  3"^a2  +  n*.  Therefore,  the  factors  of  the  binomial  are  3*  -f  a* 
and  3*  —  3^  a-*  +  a*.  Since  the  dividend  is  efpial  to  the  ]>roduct 
of  the  divisor  and  quotient,  729  +  (i«  =  (9  +  a^)  (81  -  9  a*  +  a<). 
Hence,  in  general, 

When  a  Binomial  is  the  Sum  of  Two  Equal  Odd  Powers  of 
Two  Numbers.      Let  the  student  supply  the  method  (See  Art.  36). 


128.  ELEMENTS  OF  ALGEBRA. 

Exercise  49. 

Factor  the  following : 

1.  d2a^  +  1;  1  +  x^;  a^  +  y^;  x^^  +  i/^. 

2.  a^  +  128;  x^  +  729  ^/^  64  2^6  +  y^ 

3.  aH^  +  2^10^10;  x^  +64/;  1000  a;3  +  1331  ^/^ 

4.  a^l8  +  yS  J    135^5  _!_  320  :i;2.    ^24  _|.  ^24, 

5.  2;-5  +  7/-5;    j;15  +  ^6.    ^5^5  +^5^5.    ^21  ^  J54, 

6.  a54  +  654.    1   4.  ^12.    ^^n  _|_  ^6m.    ^-f  _^  ^- f  ^ 

7.  ai2»  +  &9'" .  32  rj>  h^  c^  +  243  a^ ;  1024  a^  +  h^^. 

8.  64  a;6  +  729  a^  ;  yig  a^  +  g\  je  ;  (^2  _  J  c)^  +  8  h^c^. 

58.    Example  1.     Factor  25  a;^  -  64  2/*. 

Solution.  The  square  root  of  the  first  term  is  5  x^,  and  of  the 
last  term  8  y.  Hence,  since  the  difference  of  the  squares  of  two  num- 
bers is  equal  to  the  product  of  the  sum  and  difference  of  the  numbers 
(Art.  26),  25  a;2  -  64  3/2  =  (5  x  +  8  ?/)  (5  a;  -  8  y). 

Example  2.     Factor  {6  a  -  4)^  -  (3a  +  4x  -  4)2. 

Solution.  The  square  root  of  each  term  of  the  binomial  is  5  a  —  4 
and  3  a  +  4  ic  —  4.  Adding  the  results  for  the  first  factor,  we  have 
8  a  4-  4  X  —  8,  or  4  (2  a  +  a;  —  2).  Subtracting  the  second  result  from 
the  first  for  the  second  factor,  we  have  2a  —  4x,  ov  2  (a  —  2  a:). 
Hence,  the  factors  are  4  (2  a  +  a:  —  2)  and  2  (a  —  2x). 

Process. 

(5a-4)2-(3a-f-4a;-4)2  =  [(5  a-4)+(3  a-|-4  .'c-4)]  [(5  a-4)-(3  a+4  x-4)] 
=  [5a-4  +  3a  +  4a;-4][5a-4-3a-4a;  +  4] 
=  [8a  +  4a;-8][2a-4a:] 
=  4[2a  +  a:-2][2(a-2a:)] 
=  8(2a  +  a:-2)  (a-2x). 


FACTORING.  129 

A  binomial  expressing  the  diiference  between  two  e<iUttl  even 
powere  of  two  numbers  may  often  be  separated  into  several  factors. 
Thus, 

Example  3.    Factor  a"  —  i/". 

Solution.  The  square  root  of  each  term  of  the  binomial  is  3*  and 
y*.  Adding  these  results  for  the  first  factor,  we  have  x^  -}-  y^.  Sub- 
tracting the  second  result  from  the  first  for  the  second  factor,  we  have 
x^  —  y*.  Similarly  the  factors  of  x*  —  y*  are  ar*  +  y*  and  x*  —  y^.  In 
the  same  way  the  factors  of  z*  —  //*  aie  x^  +  y^  and  x^  —  y"^.  Finally 
the  factors  of  x^  —  j/*  are  x  +  y  and  x  —  y.  Hence,  the  factors  of  the 
binomial  are  x*  +  y*,  a:*  -I-  y*,  x*  +  y\  x  +  y,  and  x  —  y. 

Process,     x^' — y^^  =  (x^ + y^)  (x^  —  y^) 

=  (x^+y^)(x^  +  y*)(x*-y*) 

=  (x^  +  y^)  (a:*  +  /)  (x2  +  y2)  (x2  -1/2) 

=  (a:«+/)  {x*i-y*)  (xH/)  (x  +  y)  (x-y). 

Hence,  in  general, 

When  a  Binomial  is  the  Difference  of  Two  Equal  Even 

Powers  of  Two  "Numbers.  Find  tlie  square  root  of  eacli  term  of 
the  biuouiial ;  add  the  results  for  one  factor,  and  subtract  the  second 
result  from  the  first  for  the  other. 

Notes :   1.  Tlie  preceding  method  is  a  direct  consequence  of  Art.  26. 

2.  The  above  method  finds  a  practical  application  when  it  is  necessary  to 
find  the  difference  l)etween  the  squares  of  two  numerical  numbers.    Thus, 

(235)«  -  '219)2  =  (235  +  219)  (235  -  219)  =  454  X  16  =  7264. 

Exercise  50. 
Factor  the  following : 

1.  a^x^-lP-  f  ;  10  .r2  -  0  ^y2  .  25  a^x^  -  49  JV- 

2.  a,-*  -  ?/ ;  2r4  -  81  / ;  .x-^  -  ?/  ;  x^^  -  f. 

3.  a86*-81a^?/«;  l-100aH*c2;  16  a^^  -  9  6«. 

4.  9  «2'.  -  4  a^-  ;  i  «2  -  J  /,2  .  .^1  __  yl^ 

9 


130  ELEMENTS   OF  ALGEBRA. 

5.  x-^  -  /;  (^  +  hf  -  (c  +  df-  {X  -  yf  -  a\ 

6.  a^-[x-yf'Axy^a  hf  -I)  {a  ■\- hf  -  {a --  hf. 

7.  (a  +  l)2-(&+l)2;Xa+ir— C^^-l)2;  (753)2 -(253)2. 

8.  (24  X  +  yf  -{2Sx-  yf;  (1811)2  _  (689)2. 

9.  {5x-  2)2  -(x-  4)2;  (1639)2  -  (269)2. 


10.  a^"-  1;  729x'^y-  xy^;  aH  -  b^ ;  a  -  h. 

11.  (2x+  a-  Sy  -(3-2xf;  64:X-^  -  729  y-^ 

12.  (575)2  -  (425)2 ;  2  a  -  4  2:2 ;  25  a"  -  3  6''" ;  a:^  -  3/6 

59.  Compound  expressions  can  often  be  expressed  as  the  differ- 
ence of  two  equal  even  powers  of  two  numbers,  and  then  factored  by 
the  foregoing  principles.  In  many  such  expressions  it  will  be  neces- 
sary to  rearrange,  group,  and  factor  the  terms  separately.     Thus, 

Example  L     Factor  x^  -  y"^  -i-  a^  -  b^  +  2  ax  -  2h y. 

Process. 

*2_y«-|-a2-6H2ar-2&i/=:a:2+  2  ax  +  a^  -  ¥  -  2hy  -  y^ 

=  (x2  +  2  a  a:  +  a2)  -  (62  +  2  6  y  +  t/2) 
=  (X  +  a)2  -(b  +  yy 
=  [(X  +  a)  +  (^  +  b)]  [(x  +  a)-(h  +  y)] 
=  [a  +  b  -i-  X  +  y][a  —  b  +  X  -  y'\ 

Explanation.  Rearranging  and  grouping  the  terms,  in  order  to 
form  the  difference  of  two  perfect  squares,  we  have  the  third  expres- 
sion. Factoring  the  third  expression  gives  the  fourth  expression. 
The  square  root  of  each  term  of  the  fourth  expression  is  (x  +  a) 
and  (y  +  b).  Adding  these  results  for  the  first  factor,  we  have 
a  +  b  +  X  +  y.  Subtracting  the  second  result  from  the  first,  we 
have  a  —  b  +  X  —  y,  for  the  second  factor. 


FACTORING.  131 

Example  2      Factor  2xy  +  I  -  x^  —  y^. 

Process.     2zy  +  I  -  x"^  -  y*  =  I  -  x^  +  2 xy  -  y^ 

=  1  -(x2-  2x1/4-2/2) 
=  1-  (x-yY        Art  55. 

=  [l  +  (^-2/)][l-(^-l/)] 

=  [l  +  x-y][l-x  +  yl 

Example  3      Factor  4a^b^i-4c^d^^8ahcd-(a^-\^b^-c^-(P)'' 

Process.     4  a*  />»  +  4  c2  </«  +  8  a  6  c  </  -  (a^  +  />'  -  c^  -  d^y 
=  4a^b'^  -\-8abcd  +  4  c^d^  -  (a«  +  b^  -  c^  -  d'^y 
=  (2a6  +  2crf)2-  (a2 -I- 62  -  c2  -  rf2)a 

=  [(2a6  +  2c</)  +  (a2-f-6*-c2~rf''')][(2aft+2c(/)-(a2  +  62-c2-f7i)] 
=  [2a6  -f-  2cd  +  a*  +  fc'*  -  c*  -  «/2J  [2a6  +  2c</  -  a^  -  &«  +  c^  +  d^] 
=  [(a2  +  2a6  +  6-')-(c^-2crf  +  rf2)][(c2+2crf  +  f/2)_(rt2_2a6+/^'^)J 
=  [(a  +  ^y  -  (c  -  rf)2]  [(c  +  dy  -  {a  -  6)2] 

=  [(a+6)  +  (c-rf)]  [(a+6)  -  (c-^)]  [(c+rf)  +  (a-6)]  [(c+d)-(a-6)] 
=  [a  +  6  +  c  -  (/]  [a  +  6  -  c  +  dj  [a  -  6  -}-  c  +  </]  [6  +  c  +  rf  -  a]. 

Explanation.  Arranging  and  factoring  the  first  three  terms,  we 
have  the  third  expres.sion.  The  f*quare  root  of  each  term  of  the 
third  expression  is  2ab  +  2cd  and  a"^  +  1^  —  c^  —  d^  Adding  and 
subtracting  these  results,  respectively,  gives  the  fifth  expreasion. 
Rearranging  (2a  6  and  2  erf  suggest  the  proper  anangenient)  and 
grouping  the.se  terms,  gives  the  sixth  expression.  Factoring  the 
terms  of  the  sixth  expression,  we  have  the  seventh  expression 
Finally,  factoring  the  terms  of  the  seventh  expression,  we  obtain 
the  result 

Exercise  51. 
Factor  the  following  expressions : 

1.  a2-62-c2-2&c;  a^ -{- y'^ -V^ -2 a y •  l^-a^-V^^2ah. 

2.  25a:2_2^_(56c_9c2;   2?Jf.2ax-\- a^- y^- 2yz-z\ 

3.  4a;2_i2a:y+ 97/2-81;  T^-^x^Uf,  4r*-l+r)r-9z2. 

4.  16^*-  r2-f  -  -  J      9rt2_6^4-l-a:2_8a;y-16y2. 


132  ELEMENTS   OF  ALGEBRA. 

5.  x^  —  li^  +  '^'^^  —  n^—  2mx  —  2ny\  a*  +  b"^  —  c^  —  cf^ 

6.  12  xrj  -4  x^  -^J  y^  +  z^;   A  x  -  1  -  4=  x^  +  4  a\ 

7.  (^2  _  2/2  _  ^2)2  _  4  2/2;22  .    (^2«  ^  ^2"  _  c4m)2  _  4  ^,2«52« 

8.  4  ^2  _  12  a  a;  -  6-2  -  fZ2  -  2  c  c^  +  9  a2 .  4  a:^  +  9  ?/2 
-16^2  -25  6^2  -  12^?/  -  40  ^^s;  Ax^  -  W'  -2hc-c\ 

9.  :i'4  -  25  ««  +  8  a2a;2  -  9  +  30  a^  +  16  a*  ;  2/2  +  6  6aj 

-  9  62^2  _  10  ^)  ?/  -  1  +  25  ^>2 ;   (a4«  -  4  a2"  -  6)2  -  36. 

10.  rz;2«  _  9  ^2  +  ^2-  _  2  ^"^"^  _  6  a  &  -  ^>2. 

11.  a;6"-4?/*'"  +  12?/2'"^  +  2a3:i.^«_  9:^2  4.  ^6_ 

12.  4  ^2  _  9  2/2  +  16  ^2  _  36  ^2  _  16  ^ ^  +  36  ny. 

13.  tt2«  -f-  ^>2«  _  2  a*^^)'*  -  c2'»  ^  Aj^"*  -  2  c"*  A;2m^ 

14.  4  ^2  +  9  ^.2  _  16  (2/2  +  4  ^2)  _  4  (16  2/;2  _  3  «^.). 

15.  a2_,_^5_9^,2  4.i^2.  a^-a2-9-2a2^,2  +  ^4_^6a. 

60.    The  method  for  factoring  a  trinomial  consisting  of  two  trino- 
mial factors  depends  upon  the  following  axiom  : 

5.   If  the  same  numher  he  both  added  to  and  subtracted 
from  another^  the  value  of  the  latter  will  not  be  changed. 

Example  1.     Factor  x^  +  a^x^  i-  a*. 

Solution.     Adding  and  subtracting  a^x"^,  we  have  x*  +  2a^x^i-a* 

-  a^x^.  Factoring  the  first  three  terms  of  this  expression,  we  get, 
{x^  +  a2)2  —  a^x^.  Here  we  have  the  difference  of  two  equal  even 
powers  of  two  expressions,  and  it  is  equal  to  the  product  of  the 
sum  and  difference  of  their  square  roots.  Hence,  the  factors  are 
a*  +  a  a:  +  a:^  and  a^  —  ax  +  x^ 


FACTORING.  133 

Process. 

=  X*  i-  2a2x2  +  a*-a«z« 

=  {x^  +  a^y-a^x* 

=  {xHa^-a  x)  (xHa'^-a  x),  or  (aHa  x+x«)  (a«-a  z+z«). 

Example  2.     Factor  16  a*  -  17  a"  6^  +  6*.  \ 

Process.    I6a*-na^b^-^b*  =  \6a*-n  a^b^+9a^b^-hh*-9a^b^ 
=  16a<-8a«fc2  +  M-9a262 
=  (4  a^- 62)2- (3  a  6)-^ 
=  (4a2  +  3a6-6«)(4a2-3a6-Z>2) 
=  (rt  +  6)  (4  a  -  6)  (a  -  ^)  (4  a  -  6)v 

Explanation  Adding  and  subtracting  9  a*  6*  to  the  expression 
(to  form  a  perfect  square),  arranging  and  factoring  the  terms,  we 
have  the  fourth  expression  (the  difference  of  two  equal  even  powers). 
Factoring  the  fourth  expression,  we  get  the  fifth  expression.  The 
factors  of  4  a'  +  3  a 6  -  6*  are  a  +  A  and  4a  -  b.  The  factors  of 
4  a*  —  3  a  ^  —  ^^  are  a  —  b  and  4  a  +  b.     Hence, 

When  a  Trinomial  is  the  Prodnct  of  Two  Trinomial  Factors. 

Make  the  trinomial  a  perfect  scjuare  by  adding  the  requisite  expres- 
sion. Also  indicate  the  subtraction  of  the  same  expression.  The 
resulting  expre8.sion  will  be  the  difference  of  two  squares.  Take 
the  sum  of  their  scjuare  roots  for  one  factor,  and  their  diflerence  for 
the  other. 

Exercise  62. 

Factor  the  following  expressions : 

1.  9a*  +  3a2624.4fe*;  aHOa^  +  Sl;  16  X* -\- 4  aP  f  ■\- 1/*. 

2.  a^  +  a^iZ-^-f;  Sla^2Sa^a^-\-Ua^;  mHm^wa  +  w*. 

3.  40^+8^*2/^+9?/*;  a8  +  a*fe2  +  54.  81a*  +  36  aHie. 
•1    25a^-9a^}^-\'l6b^;  x^  +  xy-hy^;  a^  +  a^f  +  f. 

5.   16a8  +  8a*&3+92>«;  9a*'{'38aH^+49b^;  p^  +  pHh 


134  ELEMENTS  OF   ALGEBRA. 

6.  49  a^  +  110  a?lP'  +  81  &^   9  ^*  +  21  ^2 ^^  +  25  /. 

7.  m*"  +  m2»  +  1 ;   ^*"  +  16  :2;2«  +  256. 

8.  a2-3a&+&2;  «4«_6^2nj2m_^j4m.  25m4-44mV+16?i*. 


61.  Frequently  the  terms  of  an  expression  can  be  grouped  so  as 
to  show  a  common  factor.     Thus, 

Example  1 .     Factor  2am  +  ^hm  —  cm  —  A  an—  Qhn  +  2  en. 

Process.     2am  +  36m-cm  —  4an  —  66n  +  2cn 

=  (2  a m  —  4  a 7i)  4-  (3  & m  —  6 6 ?i)  —  (cm  —  2cn) 
=  2a  (m—  2n)  -h  Sb(m  —  2n)  —  c  (m  —  2n) 
=  (m  -  2  n)  (2  a  +  3  6  -  c). 

Explanation.  Grouping  the  terms  of  the  given  expression  in 
pairs  ;  taking  the  common  factor  2  a  out  of  the  first,  3  b  out  of  the 
second,  and  c  out  of  the  third,  we  have  the  third  expression.  Divid- 
ing the  third  expression  by  m  —  2n  (the  common  factor),  we  have 

2  a  4-  3  &  —  c.     Hence,  the  factors  are  m  —  2n  and  2  a  +  3  6  —  c. 

Example  2.     Factor  12  a^  -  4  a^b  -  S  a x^  +  b x^. 
Process. 

12a^-  4 a^b  -3  ax^  +  bx^  =  (12  a^-Sax^)  -  (4a^b-b  x^) 
=  3  a  (4a'^  -  x^)  -  h  (4  a^  -  x^) 
=  (4  a2  -  x^)  (3a-b) 
=  l2a  +  x)(2a-  x)  (3  a-  b). 

Explanation.     Grouping  the  terms  in  pairs  ;  taking  the   factor 

3  a  out  of  the  first,  and  b  out  of  the  second,  we  get  the  third  expres- 
sion. Dividing  this  by  4  a^  —  x^,  we  have  3a  —  b.  The  factors  of 
4a^—  x^  are  2  a  +  x  and  2a  —  x.  Hence,  the  factors  of  the  poly- 
nomial are  2a  +  x,  2a  —  x,  and  3  a  —  b. 

Example  3.     Factor  2mn  —  2nx  —  my  +  xy  -{-  2  n^  —  ny. 
Process.     2mn  —  2nx  —  my  +  xy  +  2n^  —  ny 

=  (2mn  —  2nx  +  2  n^)  —  (my  —  xy  +  7iy)  , 

=  2n  (m  —  X  +  n)  —  y  (m  —  X  +  n) 

=  (m  —  X  +  n)  (2  n  —  y). 


FACTORING.  136 

Example  4.     Factor  -4ax  +  4x^  +  4ay-\-4y^-8xy. 

Process.    —4ax-^4x^4ay+4y^-6xy  =  4[-ax+x^-\-ay+y^-2xy] 

=  4[{x*-2xy-{-y^)-{ax-ay)] 
=  4[{x-y){x-y)-a{x-y)] 
=  4(x-y)[(x^y)-a] 
=  4{x-y)[x-y-a]. 

Example  5.    Factor  2am^-2an^-2am-2an-\-2a^2a^4a^n. 

Solution.  Remonng  the  common  factor  2  a,  we  have  w*  —  n* 
-7n  —  n  +  a  —  a*4-2an.  Arrange  the  terms  of  this  expression  into 
the  groups  m*  —  (n*  —  2  a  n  +  a*),  and  —  (m  +  n  —  a).  The  factors 
of  the  first  group  are  m  +  n  —  a  and  m  —  n  +  a.  Hence,  m^  —  n' 
—  m  —  n  +  a  —  a2-|-2an  =  m*—  (n*  —  2an4-a*)  —  (m  +  n  —  a) 
=  (m  +  «  —  a)  (m  —  n  +  a)  —  (m  +  n  —  a).  Dividing  this  expres- 
sion by  the  common  factor,  m  +  n  —  a,  we  have  m  —  n  +  a  —  1. 
Hence,  the  factors  of  the  polynomial  are  2  a,  m  +  n  —  a^  and 
m-n-\-a-  1.  Therefore,  2am^-2an^-2am-2an'{-2a^ 
~  2  a*  -h  4  a^  n  =  2  a  (m  +  n  -  a)  (m  -  n  +  a  —  I). 

Process.     2  am^  -  2  a  n^  -  2  am  -  2  an  +  2  a^  -  2  a*  +  4 a^n 
=  2  a[m^  -  n^  -  m  -  n  +  a  -  a^  +  2 an] 
=  2a [(my  -  (n  -  a)^ -(m  +  n-  a)] 
=  2a[(m  +  n  -  a)  (m  —  n  i-  a)  —  (m  +  n  -  a)] 

=:  2  a  (m  -f  n  —  a)  [m  —  n  +  «  —  1] .     Hence, 

To  Factor  a  Polynomial  by  Gronping  its  Terms.  Group  the 
terms  of  the  polynomial  so  that  each  group  shall  contain  the  same 
compound  factor.  Factor  each  group  and  divide  the  result  by  the 
compound  factor.  The  divisor  and  quotient  will  be  the  required 
factors.  If  the  polynomial  has  a  common  simple  factor,  remove  it 
first. 

Note.  It  is  immaterial  what  terms  are  taken  for  the  different  groups  so  that 
each  group  contains  a  common  factor.  Tf  the  groups  are  suitably  chosen  the 
result  will  always  be  the  same,  although  the  order  of  the  factors  may  be 
changed.    Thus,  in  Example  3,  by  a  different  grouping  of  the  terms,  we  have 

2mn  —  2nx-my  +  xi/  +  2n^  —  ny 

=  (2mn  —  my)  —  (2nz  —  xy)-{-(2n*  —  ny) 
=  w  (2  n  -  y)  -  X  (2  n  -  y)  +  n  (2n  -  y) 
=  (2n-y)(m-x  +  n). 


136  ELEMENTS  OF  ALGEBRA. 

Exercise  63. 

Factor  the  following : 

1.  a^  ■{-  ab  -\-  ac  +  be;    a^c^  +  acd  —  '2abc  —  2bd. 

2.  am—bm—an+bn;  4:ax—ay—4:bx+by;  af^+a^-{-cr-\-a. 

3.  Qax  —  Sbx  —  Qay  +  3by;  pr  -\-  qr  —  2^  s  —  qs. 

4.  ax— 2hx-\-2by-\-4tGy  —  4:cx—  ay. 

5.  5a2- 5&2_2a  +  2&;    ^  x^  +  Z  xy  -  2  ax  -  ay. 

6.  2x^-3^^-4cX-2;  a'^x^-a^x^-a'^x^+1;  mx-2my 
—  nx  +  2  7iy\    4:  X  —  a X  -\-  4:  a  —  a^. 

7.  x^  +  mxy—4:xy  —  4:my'^\  4:a^  +  4:x'^  +  5a  — 5x  —  Sax. 

8.  3a'^—Sac  —  ab  +  bc;  a'^ x  +  ab x  +  a c  +  ab y  +  I'^y -{- b c. 

9.  ^  ax^  +  3  a  xy—5bxy—'Sby'^;  mn  + np  —  mp  —  n^. 

10.  rii^  +  np  —  mf  —  7i?\  V^y^—2 x^y  +  ?>a^—21  xy^. 

11.  21  a- 5c  +  3  «c- 2&C- 14  &- 35;  i^2_5^2/ 
+  62/^+32:  —  6?/;   2:^  —  r?;2-|-2;_l, 

62.    Example  L     Factor  a;^  +  1/2  +  g'-J  -  2  a;  1/  +  2  a;  2  -  2  ?/  2. 

Solution.  The  expression  consists  of  three  squares  and  three 
double  products.  Hence,  it  is  the  square  of  a  trinomial  which  has  x, 
y,  and  z  for  its  terms.  Since  the  sign  of  2  x  2  is  + ,  and  2xyis—, 
X  and  z  have  like  signs,  while  x  and  y  have  unlike  signs.  Hence, 
one  of  the  two  equal  factors  is  x  —  y  +  z. 

.-.  x^  +  y^  +  z"^  -  2  xy  +  2  xz  -  2yz=  (x  -  y  +  z)^. 

Example  2.     Factor  x^  -  3  x^  y  -\-  3  x  y^  -  y^. 

Solution.  It  is  seen  at  a  glance  that  the  given  polynomial  fulfils 
the  laws  stated  in  Art.  29.  Therefore,  one  of  the  three  equal  factors 
is  x  -  y.     .'.  x^  —  3  x^ y  +  3  X  y^  —  y^  =  (x  —  yy.     Hence, 


FACTORING.  137 

When  a  Polynomial  is  a  Perfect  Power  of  an  Expression. 

By  observing  the  exponents,  coetticients,  and  signs  of  the  terms,  find 
such  expression,  as  raised  to  a  given  power,  will  produce  the  polyno- 
mial.    This  expression  will  be  one  of  the  equal  factors. 

Exercise  64. 

Factor  the  following : 

1.  a^  -f  2  a  5  +  Z>2  +  2  a  6-  +  2  6  c  +  c2 

2.  a2-2a6  +  62_2«c-i-26c  +  c2. 

3.  a^+h^-\-c^-\-2ab-2ac-2bc;  16-f  32a:  + 24ic2 

4.  a^-15a*x  +  90a^x^- 2433:^ -  270a^JT'^+  405aa^. 

5.  a^-2ab  +  }r^+2ac  +  c^-2ad-2bc  +  (l^-2cd-i-2bd. 

6.  27x^i/-l08a?x^y^-64:a^+lUa^xy, 

7.  m^—2p  x—2  n  x-\-n^+]f—2  mn+2m  x-\-3?—2  mp+2  np. 

Miscellaneous  Exercise  55. 
Factor  the  following : 

Vott.    If  the  expression  has  a  common  simple  factor,  it  should  be  first 
removed. 

1.  10ar»"-30a:"-40;  ar^  +  A^+l;  \22^if-Z(Sxy-A8. 

2.  a;2  _  .56  a;  +  .03  ;  a2  +  f  I  a  +  1  ;   ^  ^  x  -  x^. 

3.  3m«n8-3m*7i;  16^8-2;  a^-Sl;  6a;54-48aJ*+72a^. 

4.  ar»ya-5^^y-^j;  aH2-|^a3J-^;  9(a  +  J)2- 
+  ^xy{a  +  by-x^f. 


138  ELEMENTS  OF  ALGEBRA. 

5.  a'^—2ax—4:a  +  x^+4:x;  8— 2x^—4:0^— 2x^;  da^+a. 

6.  x^  +  if  ^^  -  3%;    ^2«  +  16  a"  +  63;     -i-|  a^'"  + 
(f  ^t"  -  -¥-  ^^")  a^'"  -  6  a**  +^. 

7.  m^  —  a  m  —  71  m  +  a  71 ;  a^  +    7  a  —  8  ;  4  a^  —  4  2?^ 
-2a  +  2b;  49  a^H^""  +  7  (:i;2  +  32/^)a"53«^i  _^  3iz;i2/i 

8.  204-5a-a2;    ^8"  - -L2_8  ^4n  ^  1|^ 

9.  m^  —  n^  —  mp  —  np  ;  a;^  —  ic^  +  a;*  —  2^^ ;  a^  —  a^  &^. 

10.  380-^-^2.    8  ^10".  _  9  ^5m  _|.  1 .    (^^  _  ^)4« 

11.  6a.^2_2,__  77.    12  2:2+108^+168;    ^2^2^^^/ 
+  2/2  —  5^—5?/;  I  2^2  —  Jg-  (5  -m  71  +  3  2/)  ^  +  ^z'^^^'^  V- 

12.  l-Tiy^-Tfy^^  2  2^2+ 5  ^?/- 3?/2-4aa;  +  2a7/. 

13.  a2«  +  (^  _j_  ^)  ^«^«  +  ^/2"  ^7/;  :r2  +  (a  +  &)  ^  -  2  a2 

-  a  & ;  a;- 12  -  7/^ ;.  81  x^  -  22  2^2 ^2  +  y^^ 

14.  al2&  +  Z;13.    ^3  +  ^3_^3^^(^_|_^).    a:4'»+c(a  +  5)2:2« 

—  ah  {a  —  c)  (b  +  c)  ]   m^  +  71^  +  771  +  71. 

15.  a2_2,2_c24.^/2_2(^^_2,c);  4  +  4^'+2«y+^2_^2_2^2^ 

16.  x^-V{a  +  2h)x-\-ah-^h'^;  x^^''+(a-b)x^''-2a'^-2ah. 

17.  250  (m  -nf±2;  8  (771  +  nf  ±  (2  m  -  n^. 

18.  52»c2»2;2'»_     6"c"a2x"    -&~C"2:"+ft2;   49^4_i5^2^2 

+  121  2* ;   (7?i  +  nf  ±  (m.  -  nf. 

19.  4  (771  •—  n)^  —  (m  —  n);  {m  -{■  Tif  ±  m  {m  +  n). 


FACTORING.  139 

20.  x^-b{a-c)x-ac{a-\-b)(b  +  c);  64:m^-{-12Sm^n^ 

21.  6  2^  +  rS3^i/ +  6xf-  6x^f-xi/-12f. 

22.  2^'*  -{She  +  ac  +  ab)x''  i-  'Sabc(b  ■{■  c). 

23.  m3  +  4  m  /i2  ±  8  n^  ±  2  m^  n ;  {711  +  3  n)2  -  9  (m  -pf. 

24.  x^''  +  (a  -\-  b  -c)x^^''-  ac  -  be;    {x -^  yf  -  x 

-  y  -  6  ;  25  ^  +  24  x^i/  +  16  f, 

25.  9a:*y*— 3  2^^—60,-2 y^;  m^— mri  — 6  71^^47/1  ip  12  7^. 

26.  rrAn^{ab-zf-m^n^{xy^-2zf\  81a2"_i99a»fe'» 
+  121  62-;   81  a^"  -  99  a2«^,4«  4.  25  fts™ 

27.  18  a:2  _  24  xy  +  8  /  ±  36  a;  ^  24  v/ ;  2  m2  j^  2mn 

-  12  ?i2  -  12  am  -  36  a 71 ;  a^a^  ±  64  a?iA 

28.  2^^+  3x'y-282/*+  28y  +  4a:;  2y  -  ^ay  -\- 4.bx 
-\-  G  a  X  —  2x  —4by. 

29.  7W*  71  —  7?l2?i3  —  7;i3^2  _|_  ^^^  ,^4.   ^j^4  _  ^^^^  ^  ^^4 

30.  15  a^J  -  16  7/2  -  15  a  a:  -  8  a;  y  +  20  a  y ;   (a  -  6) 
(a2  _  c2)  -  (a  -  c)  (a2  -  ^2). 

31.  c«^-c2-a2c3rf3  4.rt2;  m87l3±512;247n,27l2-3077l7l3 

—  36  71*;  aa^  —  3bxy  —  axy-\-3b/. 

32.  7m2- 7n7i- 69124. 16m -327i;  4  a;^  +  4 ajS -  a;2  -  ar*. 

33.  4  m8  -  4  7i3  _  3  71  (712  _  m^)  +  2  7?i  (71  -  7?i)2 

34    9m»±9a2m7;  a.^~16y2  +  a;±42^:  (x-2xy)^ 

—  (a;  —  2  a;  y)  —  6.     Query.     How  many  factors  in  the  first  part  ? 


140  ELEMENTS  OF  ALGEBRA. 

35.  64.(4:  x  +  yf  -  49  (2 a;  -  3  7/)2 ;  (wt* -  iii^ -  5)2  -  25. 

36.  m^  +  m^  7i  +  m  7i^  +  iii^  ii^  +  m^  n^  +  rn?  n^ ;  (a;  —  y)^ 

-  1  +  2:y  (a;  -  ?/  +  1) ;  (^2  +  4)2-16  a:2. 

37.  (m2  +  3  m)2  -  14  (m2  +  3  ??0  +  40  ;   (m  7i  -    t!-)^ 

-  m  ?i  (tw  %  —  72.  —  3)  —  9. 

38.      :C2H  _  ^n  _  I   ^   ^-n   ^   ^-2n.    ^-f  _  ^-f^ 

39.  14^2 ^3_  35a3  2;2+  14a*:r;  a;-6 -t/"! 

40.  12:^:5-8^3^2+  21  ^2^/;  64:cl±  27 ici 
Separate  into  four  factors : 

41.  {x-2y)o(^-{y-2  x)t/;  (^'"+ 6^"*  + 7)2- (^'"+ 3)2. 

42.  4  ^2  (^3  +  18  a  &2)  _  (32  a^  +  9  ^2  a?)  ■      iQ  ^^  ,^2 

-  (m2  +  4  ?^2  -  ^2)2  .      (^4m  _  2  rt2-^,2n  _  J4«)2  _  4  ^4m  j4» 

■    43.   x^  +  ^./  -  8  0^6  ^3  _  g  ^9 .  ^9m  _^  ^sm  +  54  _^3»t  +  64 

44.  m^  —  2  (?t2  +  ^2^  ^.^2  _|.  (^2  _  ^2^2 

Separate  into  five  factors  : 

45.  m''  —  inP  n^+  2  7?i*  n^  —m^n'^;   6  m*  7^2  +  m^  n  —  6  m^  n^ 

-rn^n^l      (a;2m  _^  ^2n  _  20)2  _  (^^'^^n  _  y2n  _^  ]^2)2 

46.  ^7'«  +  ^4/n_;j^g^m__]^g.     16  ^7m_81  ^  »t_  ^g  ^4m_^  3]^ 

Separate  into  seven  factors : 

47,  ^12m_^8mj4n_^4mj8n_^   J12n^ 


HIGHEST  COMMON  FACTOR  141 


CHAPTER  XII. 

HIGHEST  COMMON   FACTOR. 

63.  The  product  of  any  of  the  factors  of  a  number  is  a 
factor  of  the  given  number. 

Thu8,  since  30  =  2  X  3  X  5,  6,  10,  and  15  are  factors  of  30. 

The  product  of  the  common  factors  of  two  or  more  num- 
bers must  be  a  factor  of  each. 

Thus,  since  42  =  2  X  3  X  7,  and  66  =  2  X  3  X  11,  2  X  3,  or 
6,  is  a  factor  of  42  and  66. 

The  product  of  the  higliest  powers  of  all  the  factors  which 
are  common  to  two  or  more  numbers  must  be  the  greatest 
common  factor  of  the  given  numbers. 

Thus,  since  24  =  2»  X  3  and  36  =  2^  X  3^,  2^  X  3,  or  12,  is  the 
greatest  common  factor  of  24  and  36. 

The  Highest  Common  Factor  (H.  C.  F.)  of  two  or  more 
algebraic  expressions  is  the  expression  of  highest  degree 
which  will  divide  each  of  them  exactly. 

Thus,  3a:«y«i8the  H.C.F.  of  3a:«j/«,  Qx^y\  and  15x<y«2. 

Note  1.  Two  or  more  exprcRsions  are  said  to  be  prime  to  each  other  when 
they  have  no  common  factor.    Thus,  5  a*  and  9  b  are  prime  to  each  other. 

Example  1.  Find  the  H.  C.  F.  of  24  a«  6«  c«,  60  a'^'c^y^ 
48a»62c«,  and  36a2  6*c«x». 


142  ELEMENTS  OF  ALGEBRA. 

Process.     24  a^b^c^  =  2^  X  3    X  a^  X  b^  X  c^  ; 

60  a^b^c^f  =  2^  X  3    X  5  X  a^  x  6^  x  c^  X  2^2  ; 
48  a^b^c^  =  2*  X  3    X  a^  X  b^  X  c^ ; 

36  aH^c^x^  =  22  X  32  X  a2  X  6^  x  c^  X  x^ 

.'.  theH.C.F.  =  22  X  3  X  rt2  X  62  X  c2=:  12a2Z>2c2. 

Explanation.  Factoring  each  expression,  it  is  seen  that  the  only- 
factors  common  to  each  are  22,  3,  a^,  b\  and  c^.  Hence,  all  of  these 
expressions  can  be  divided  by  any  of  these  factors,  or  by  their 
product,  and  by  no  other  expression. 

Example  2.  Find  the  H.  C.  F.  of  2  x^  -  2  a:  3/2,  4  x^  -  Axy^,  and 
2  x*  -  2  a:2  3/2  +  2  x3  3/  -  2  X  ?/3 

Process.     2  a:^  —  2  x  3/2  =  2  x  (x  +  y)    {x  —  y)  \ 

4 x^  -  4  xy*  =  2^ X  (x  +  y)    {x  -  y)  (a;2  +  y^)  ; 
2  x^ -2x'^y^+  2  x^ y -2xy^  =  2  x  (x  +  y)^  (x  -  y); 

.-.  the  H.  C.  F.  =  2  a:  (a;  +  ?/)  (a:  -  2/)  =  2  a;  (a:2  _  y^). 

Explanation.  Factoring  each  expression  it  is  seen  that  the  only 
factors  common  to  each  are  2,  x,  x  +  y,  and  x  —  y.  Hence,  all  of 
these  expressions  can  be  divided  by  any  of  these  factors,  or  by  their 
product,  and  by  no  other  expression. 

Note  2.  If  the  expressions  contain  different  powers  of  the  same  factor,  the 
H.  C.  F.  must  contain  the  highest  power  of  the  factor  which  is  common  to  all 
of  the  given  expressions. 

Example  3.  Find  the  H.  C.  F.  of  8a^  x^  +  IQa^x^  +  8  a^  x\ 
2  a*  a;2  -  4  a^  X  -  6  a6,  6  (a^  +  a  xy,  and  24  (a2  a;  +  a  x^y. 

Process. 

8a5x2+16a4x3+8a8x4=      23X        a^Xx^  (a  +  xY; 

2a^x^-4a^x-6a^=  -2  X        a'^X       (a  +  a;)  (3a-a;); 
6(a2+aa:)2=      2  X3Xa2x       (a  +  x^; 
24(a2x  +  aa:2)8=      2^  X  3  X  a^  X  x^  (a  +  a:)3. 

The  common  factors  are  2,  a^,  and  a  +  x. 
.-.  the  H.C.F.  =  2a2  (a  +  x).     Hence, 


HIGHEST  COMMON  FACTOR.  143 

To  Find  the  H.  G.  F.  of  Two  or  more  Expressions  that  can 
be  Factored  by  Inspection.  Separate  the  expressions  into  their 
factors.  Take  the  product  of  the  common  factors,  giving  to  each  fac- 
tor the  highest  power  which  is  common  to  all  the  given  expressions. 

Exercise  56. 
FindtheH.C.F.  of: 

4.  12  a^lJ^x^  and  18  a^bs^  ;  6  ci^xy,  8  aT^y,  dindi  ^tii^xy^. 

5.  loa^a^y^,  ^a^x^f,  and  21d^x^7/. 

6.  12  2:8^2  22^  18  a:*/^^,  and  36  a:^^^^. 

7.  20  c8a:V,  8a2x2yl,  and  \2a^x^yi. 

8.  a^hx  ■\-  al?x  and  a^h  —  l/^. 

9.  a^y'^  —  z^  and  ax^y  ^h  xy  -\-  axz  —  hz. 

10.  3a;*+8a:8+4ar*,  ?^a^^-ll3^+^3^,  and  Za^l^7?-\2x^. 

11.  Za^x^y-Za^xy-Z^a^y  and  3a2a:3-48  a'^ x 
-  3  a2  0^2  4.  48  a\ 

12.  x^-{-x,{x^  1)2  and  2^8  +  1 ;  a^"  +  a:"  -  30  and 
a^-  _  a:-  _  42;  a:8  ^  27,  a:^  _  9^  and  2  a^  +  5  a;  -  3. 

13.  a^  —  a^y,  a^  —  xy^,  and  7^  —  xi^. 

14.  a;*  -f  aj2  y2  _^  y4  ^nd  a^  —  2ar^y+  2xi^  —  i^. 

15.  12  (a  -  hf,  8  (a2  -  ^2)2^  and  20  (a*  -  ?>4). 

16.  8  a:  2;  (a;  —  y)  (x  —  z)  and   12yz(i/  ^  x)(y  —  z). 

17.  4ar»-  12a:  +  9,  4a^-  9,  and  4a^bx-6a^b. 

18.  a?-21f,  x^-^xy^^f,  and  2  r*- a;y~  15/. 


144  ELEMENTS   OF  ALGEBRA. 

19.  :r}'  —  if,  (x^  —  y^f,  and  ax^  —  1  axy  +  iS  ay^. 

20.  ma^  —  mx,  2x^-\-l^  x'^—2^  x,  and  4:a?x^—4:a^x. 

21.  24m7i-C?/i+16  ??,-4,  649i2-4,  and  IG^i^-S^i-f  1. 

22.  a;2  +  4  a;  +  4,  ^3  +  g,  and  4  ic2  +  2  a;  -  12. 

23.  16  ^3  _  432,  a;2  -  6  ^  +  9,  and  5  x^  -  13  rr  -  6. 

24.  rii^—n?,  m7i  —  7i^+mp—np,  and  m^—m?n  +  mn^—n^. 

25.  6  x^  —  9Q  X,  m  a^  y  —  8  m  y,  and  15  ^  a:^  —  60  ^. 

26.  a;6«-ll2;3«+30,  r^6«_i3^3«4.42^  a^i^j  ^6n_^^3n_42. 

27.  a;3''  -125,  ^2  «  _  10  ^«  +  25,  and  2  2^2«  _  ^  ^n  ^  5^ 

28.  Sa^''-  125,  4^2«_25,  and  4a:2«- 20^"  +  25. 

64.  If  the  expressions  cannot  readily  be  factored  by  inspection, 
we  adopt  a  method  analogous  to  that  used  in  arithmetic  for  the  great- 
est common  divisor  of  two  or  more  numbers.  The  method  depends 
on  two  principles  : 

1.  A  factor  of  any  expression  is  a  factor  of  any  multiple 
of  that  expression. 

Thus,  4  is  contained  in  16,  4  times;  it  is  evident  that  it  is  con- 
tained in  5  times  16,  or  80,  5  times  4,  or  20  times.     In  general, 

Since  a  factor  is  a  divisor,  if  a  represent  a  factor  of  any  expression, 
m,  so  that  a  is  contained  in  m,  b  times,  it  is  evident  that  it  is  con- 
tained in  r  m,  r  times  h,  or  r  h  times. 

2.  A  common  factor  of  any  two  expressions  is  a  factor 
of  their  sum  and  their  difference,  and  also  the  sum  and  the 
difference  of  any  multiple  of  them. 


HIGHEST  COMMON  FACTOR.  145 

Thus,  4  is  contained  in  36,  9  times,  and  in  16,  4  timed.  Hence,  it 
is  contained  in  36  +  16,  9  +  4,  or  13  times,  and  in  36  -  16,  9—  4,  or 
5  times.  Again,  4  is  contained  in  5  times  36,  5  times  9,  or  45  times; 
also,  4  is  contained  in  10  times  16,  10  times  4,  or  40  times.  Hence, 
it  is  contained  in  180  +  160,  45  +  40,  or  85  times,  and  in  180-160, 
45  -  40,  or  5  times.     In  general, 

Let  a  be  a  factor  of  m  and  n,  so  that  a  is  contained  in  m,  b  times, 
and  in  n,  c  times.  Then  (m  +  7i)  +  a  =  6  +  c  ;  also,  (m  —  n)  -{-  a 
=  b  -  c.  Again,  since  a  is  contained  in  m,  b  times,  it  is  evident  that 
it  is  containe<l  in  r  m,  r  times  b,  or  rb  times  ;  also,  since  a  is  con- 
tained in  n,  c  times,  it  is  contained  in  s  n,  «  times  c,  or  sc  times. 
Hence,  rm  -^  a=irb,  and  sn  -r  a  =  sc.  Adding  these  equations,  we 
have  {rm  -^  s  n)  ■Ta  =  rb-\-sc;  subtracting  the  second  equation 
from  the  first,  we  have  (rm  —  .<*  n)  +  a  =  rb  —  sc.  The  last  two 
et^uations  may  be  written  (r  7n  ±  s  n)  -^  a  =  r  b  J^  s  c.  Therefore, 
rm±sn  contains  the  factor  a.  , 

Example  1.  Find  the  H.C.  F.  of  4  a:«  -  3  a:^  -  24  a;  -  9  and 
8  x»  -  2  or^  -  53  X  -  39. 

Solution.  The  H.C.  F.  cannot  be  of  higher  degree  than  the  first 
expression.  If  the  first  expression  divides  8  x*—2  x^  —  b'Sx  —  39,  it  is 
the  H.C.  F.  By  trial,  we  find  a  remainder,  4  2:*  —  5  a;  -  21.  The 
H.C.  F.  of  the  given  expressions  is  also  a  divisor  of  4  x*  —  5  a:  —  21, 
because  4x*  —  5x  —  21  is  the  difference  between  8  a:*  —  2  z^  —  53a: 
-39  and  2  times  4  a:»  -  3  a;*  -  24  z  -  9  (Principle  2).  Therefore, 
the  H.C. F.  cannot  be  of  higher  degree  than  4x'  — 5z  — 21.  If 
4  z^  -  5  a:  -  21  exactly  divides  4  x«  -  3  r*  -  24  z  -  9,  it  will  be  the 
H.  C.  F.  By  trial,  we  find  a  remainder,  2  z^  -  3  z  -  9.  The 
H.C.  F.  of  4  z2  -  5  z  -  21  and  4  z«  -  3  z*  -  24  z  -  9  is  also  a  divi- 
sor of  2  z*  —  3  z  —  9,  because  2  z*  —  3  z  —  9  is  the  difference  between 
4  z»  -  3  z»  -  24  z  -  9  and  z  +  1  times  4  z^  -  5  z  -  21  (Principle  2). 
Therefore,  the  H.  C.F.  cannot  be  of  higher  degree  than  2z*  — 3z  — 9. 
If  2  z*  —  3  z  —  9  exactly  divides  4  z''  —  5  z  —  21,  it  will  l)e  the 
H.  C.  F.  By  trial,  we  find  a  remainder,  z  -  3.  The  H.  C.  F.  of 
2  z*  —  3  z  —  9  and  4  z*  —  5  z  —  21  is  also  a  divisor  of  z  —  3,  because 
z  —  3  is  the  difference  between  4z*  —  5z  —  21  and  2  times  2z*  — 3z 
—  9  (Principle  2).  Therefore,  the  H.C.  F.  cannot  be  of  higher  de- 
gree than  z  —  3.     If  z  —  3  exactly  divides  2  x*  —  3  z  —  9.  it  will  be 

10 


146  ELEMENTS  OF  ALGEBRA. 

the   H.C.F.     By  trial,  we  find  that  a:  —  3  Ib  an  exact  divisor  of 
2x^-3x-9.     Therefore  a;  -  3,  or  3  -  a;  is  the  H.  C.  F. 

Process.     4x^-3x^-  24a:  -  9  )  8a;3  ~  2a;2  -  53a;  -  39  (  2 
2  times  the  divisor,  8  a:^  —  6  x^  —  48  a;  —  1 8 


First  remainder. 

4a,2-    5a: -21 

4^2  _  5 
X  times  second  divisor, 
Second  remainder, 

X- 

-2l)4a:8-3a:2-24a:-    9(a:+l 
4a:3-5a;2-2la: 

2x2-    2x-    9 

2  times  third  divisor. 

2: 

r2-3a:-9)4x2-5a:-2l(2 
4x2-6a:~18 

Third  remainder. 

X-    3 

2  a:  times  fourth  divisor, 
Fourth  remainder, 

3  times  fourth  divisor, 

x-3)2x2_3x-9(2x  +  3 
2x2 -6x 

3x-9 
3x-9 

Therefore,  the  H.  C.  F.  =  x  -  3,  or  3  -  x. 

Note  1.  The  signs  of  all  the  terms  of  the  remainder  may  be  changed :  for  if 
an  expression  A  is  divisible  by  —  £,  it  is  divisible  by  +  B.  Hence,  in  the 
above  example,  the  H.  C.  F.  is  a:  —  3,  or  3  —  a;. 

Example  2.  Find  the  H.C.  F.  of  4x^  -  x^y  -  xy^  -  6  y^  and 
7x8  +  4x2^/  +  4X^2_3^8 

Process. 

4x8— x2i/—x2/2— 5  2/8)7x8+  4^2?/+  Axy^—  33/8(7 
4  times  first  dividend,  28x8+16x2  2/+16x?/2— 12^/8 
7  times  first  divisor,      28x8—  7x^y—  7xy^—Sby^ 
First  remainder,  23x2^+23xp+23/  =  23  ?/(x2+xi/+.y2) 

x2+xy+2/2)4x8—    x^y—    xy^—5y^(^4x-  6y 
Ax  times  second  divisor,  4x8+4x23/+4x.?/''^ 

Second  remainder,  —bx'^y  —  bxy^—by^ 

—  5y  times  second  divisor,  —bx^y  —  bxy'^—b y^ 

Therefore,  the  H.  C.  F.  =  x2  +  xi/  +  ?/. 

Explanation.  Arrange  according  to  descending  powers  of  x,  take 
for  the  divisor  the  expression  whose  highest  power  has  the  smaller 
coeflficient,  and  multiply  the  dividend  by  4  (to  avoid  fractions). 
Since  4  is  not  a  factor  of  4  x^  —  x'^y  —  xy^  —  5 


HIGHEST  COMMON  FACTOR.  147 

given  expressions  is  the  H.  C.  F.  of  4  x*  —  a;'*  y  -  xy^  —  b  y*  and  28  x« 
+  16  a^»y+ 16  a: y2- 12  i/«  (Principle  2).  Since  23  y  {x^ -{- x  y -\-  y^) 
is  the  difference  between  4  times  the  dividend  und  7  times  the 
divisor,  the  H.  C.  F.  oi  the  given  expressions  is  a  divisor  of  it 
(Principle  2).  Therefore,  the  H.C.  F.  cannot  be  of  higher  degree 
than  23y  (x^  -^  ^y  +  y^)  If  the  first  remainder  exactly  divides  the 
first  divisor,  it  will  be  the  H.C.F.  Since  23  y  is  not  a  factor  of 
the  first  divisor,  it  can  be  rejected.  Therefore,  x^  -\-  xy  +  y^  is  the 
H.C.F. 

This  method  is  used  only  to  determine  the  compound 
factor  of  the  H.  C.  F.  If  the  given  expressions  have 
simple  factors,  they  must  first  be  separated  from  them, 
and  the  H.C.F.  of  these  must  be  reserved  and  multiplied 
into  the  compound  factor  obtained.     Thus, 

Examples.  Find  the  H.C.F.  of  54x^y  +  60x'^y^  -  I8x»y* 
-  132  a:V  and  18x«ya  -  50  a^^y*  +  2x*y*-  I2x^y\ 

Solution.  Removing  the  simple  factors  6x^y  and  2x^y^j  and 
reserving  their  highest  common  factor,  2x*y,  as  forming  a  part  of  the 
H.C.F.,  we  are  to  detennine  the  compound  factor  of  9a:*  -  22x'^y^ 
-3xy*-\-l0y*  =  A  and  9x*  -  6x^y -\-  x^y^  -  25  y*  =  B.  UA 
exactly  divides  B,  it  is  the  H.C.F.  of  ^  and  B.  By  trial,  we  find 
the  remainder  -y(6x*-23x^y-  3  x !/«  +  35  .y«).  The  H.  C.  F.  of 
A  and  B  is  also  a  divisor  of  this  remainder,  because  the  remainder 
is  the  difference  between  B  and  1  times  A  (Principle  2).  Reject  -y 
from  this  remainder,  since  it  is  not  a  common  factor  of  A  and  B,  and 
represent  the  result  by  D.  The  H.C.F.  of  Z)  and  2  A  (a  multiple 
ofi4)  is  the  H.C.F.  n\  A  ami  J5  (Principle  2).  This  cannot  be  of 
higher  degree  than  D  ;  and  if  D  exactly  divides  2-4,  it  is  the  H.C  F. 
By  trial,  we  find  a  remainder,  153  y^  (3  x^  -  x  y  -  5  y^).  The 
H.C.F.  of  D  and  2^  is  also  a  divi.sor  of  this  remainder.  Reject 
153 y«,  and  represent  the  result  by  E.  The  H.C.  F.  of  E  and  D  is 
the  H.C.F.  of  D  and  2  A  ;  and  if  E  exactly  divides  D,  it  is  the 
H.C.  F.  By  trial,  we  find  that  E  is  an  exact  divisor  of  D.  There- 
fore, E  is  the  H.C.  F.  of  A  and  B.  Hence,  the  H.C.  F.  of  the  given 
expressions  is  2x^y  (3x^  -  xy  -  5  y^). 


148  ELEMENTS   OF  ALGEBRA. 

Process. 

9x^22x^y^-'3xy^+10y^  =  A  )9x*-Qx^y+     x^y'^  -25y^=B{l 

1  times  the  first  divisor,       9x*  -22xY-3xy^+10y^ 

-6  a;8  y+2S  x'Y+'^  x  2/3-35  ?/'» 
=  -y{6x^-23x^y-Zxy^+36y^) 

{3x  +  2Sy 

6x^-23x'2y-3xy^+25y^  =  D)  ISx*  -  44xY-     ^^y^+  ^Oi/*  =  2A 

3x  times  second  divisor,  lSx*-^9x^y-    9x'^y'^+l0bxy^ 

Second  remainder,  69^:^-  3bxY-^^^xy^+  20?/^ 

2  times  second  remainder,  138x3^-  1{)xY-222xy^+  40y* 
23 y  times  second  divisor,  l38x^y-529xY~  G9xy^+S05y^ 
Third  remainder,  459xV-153x2/8-7652/4 

=  I53y^(3x^-xy-5y'^) 

3x^-xy-5y^  =  E)6xf^-23x^y-   3xy^  +  35y^  =  D{2x-7 y 

2x  times  third  divisor,  6x^~   2x^y  —  10xy^ 

Fonrth  remainder,  -2lx^y+    7xy^-\-35y^ 

—  72/ times  third  divisor,     —21x'^y+    7xy^-]-35y^ 
Therefore,  K.C.F.  =  2x^y  {3x^-xy~5y'^). 

Example  4.  Find  the  H.  C.  F.  of  90  x^  y^  -  200  x^  y^  -  10  x^  y* 
and  144  x*  7/  -  64  a:  i/*  -  16  x^  /  -  144  x^  y^. 

Removing  the  simple  factors  lOx^y^  and  16  xy,  and  reserving 
their  highest  common  factor,  2xy,  as  forming  part  of  the  H.  C.  F., 
arranging  according  to  descending  powers  of  x,  we  have 

Process.     9x^—xy^—20y^ )  9a;8— Qx^i/—    xy'^—  4y^{^ 

1  times  the  first  divisor,    9^^  —     xy^—20y^ 

First  remainder,  —9x^y  +16^^  —  —y(9x^—l6y^) 

9x2-162/2)9x3-      xy^-20y^{x 
x  times  second  divisor,        9x^—l6xy^ 
Second  remainder,  1 5  x  ?/2  —  20  i/3  =  5  ^/^  (3  x  —  4 1/) 

3x-4?/)9x2-16y''=(3x  +  4i/)(3x-4  2/)(3x  +  4i/ 
3x  +  4y  times  third  divisor,  (3x  +  4y)(3x  —  4y) 

.•.  the  H. C.F.  =  2xy  (3x  —  4y).     Hence,  in  general, 


HIGHEST  COMMON  FACTOR.  149 

To  Find  the  H.C.F.  of  Two  Polynomials  that  cannot  readily 
be  Factored  by  Inspection,  if  the  given  expressions  have  simple 
factors,  remove  them  and  iirraiige  the  resulting  expressions  acconling 
to  powers  of  a  common  letter.  Take  that  expression  which  ia  of 
lower  degree  for  the  first  divisor;  or,  if  both  are  of  the  same  degree, 
that  whose  first  term  has  the  smaller  coefficient.  If  there  is  a  re- 
mainder, divide  the  first  divisor  by  it,  and  continue  to  divide  the  last 
divisor  by  the  last  remainder,  until  there  is  no  remainder.  The  last 
divisor  will  be  their  highest  common  factor.  The  highest  common 
factor  of  the  simple  factors  multiplied  by  the  last  divisor  will  give 
the  H.C.F.  sought.  " 

Notes  :  2.  If  the  first  term  of  the  dividend  or  of  auy  remainder  is  not  ex- 
actly divisible  by  the  first  term  of  the  divisor,  that  dividend  or  remainder  must 
be  multiplied  by  such  an  expression  as  will  make  the  fii-st  term  thus  divisible. 

3.  Observe  that  we  may  multiply  or  divide  either  of  the  polynomials,  or  any 
of  the  remainders  which  occur  in  the  course  of  the  work,  by  any  factor  which 
does  not  divide  hoth  of  the  polynomials,  as  such  a  factor  can  evidently  form  no 
part  of  the  H.  C.  F. 

Exercise  57. 
Find  the  highest  common  factor  of : 
1.    ar3  4-  2  ar2  -  13  a:  +  10  and  2:3  _j.  ,2  _  10  a;  +  8. 

3.  a^-x^-5x-Ssind  3^-4x^-11  x- 6. 

4  x*-93^+29x^-39x+lS  and  4 a^- 27x^+56 a: -33. 

5.  2^  —  5  ax^  4-  4  a^x  and  sd^  —  ax^  +  Za?x^  —  3  a^x, 

6.  2f-l0xf-^%x^y  and  ^x'^-^xif-^-Za^f-^s^y. 

7.  2a:S-lla;2_9  and  4a;«+  ll2:*4-81. 

8.  18  ar^  +  3  a:  -  6  and  18  a:^  _,.  95  ^  ^  104  a;  +  32. 


150 


ELEMENTS   OF  ALGEBRA. 


9.  15  m^  n^  —  20  m^n?  —  65  m^n  —  30  m^  and  2amn^ 
-\-  20  am n^  —  16  a  mn—  186  a  m. 

10.  36  m^  +  9  m3  -  27?/i*  -  18 ?7i5  and  27^57^2  -  9  m%2 
-18  m*  72.2 

11.  S  x^  —  3  X  y  +  xy^  —  y^  and  4:X^y  —  5  xy^  +  y'^. 

12.  mnx^  —  82  m  7i  a;  —  3  //i  ?^  and  m^  ti^  ^5  _|_  28  ^^5  ^2^2 

-  9  m^n\ 

13.  a;3  -  4  ^2  +  2  ^  +  3  and  2  ^*  -  9  ^3  ^  12  ^2  _  7 

14.  16  x^-^^xy  +  10  7/2  and  6  :?:*  -  29  2;3  7/  +  43  i«2  ^2 
-20^?/^. 

15.  2m^n  —  10m^n7j'^+  IS m^ny^+  224:mny^+2d4:ny^ 
and  4  m*  71  —  20  m2  ?i  7/2  —  48  in  ny^  +  112  7i  7/*. 

16.  27?2.'*a;4"-  27?i"2;3"  -  4  7??,**^2«  +  4^";:c"  and  6m"a?5n 

-  18  m^'x'^''  +    12  77l"^3«  _|.    6  ^«^2n  _  (3  ^n^n 

Query.     How  many  factors  in  this  result  1 


65.  To  Prove  the  Method  for  Finding  the  H.  C.  F.  of  any 
Two  Algebraic  Expressions.  Let  A  and  B  represent  the  ex- 
pressions, the  degree  of  A  being  either 
equal  to  or  higher  than  that  of  B.  Di- 
vide A  by  B,  and  let  the  quotient  be  m 
and  the  remainder  D ;  divide  B  by  D, 
and  let  the  quotient  be  n  and  the  re- 
mainder E  ;  divide  D  l)y  E,  and  let  the 
quotient  be  r  and  the  remainder  zero; 
that  is,  E  is  supposed  to  be  exactly  con- 
tained in  D. 

We  will  first  prove  that  -B  is  a  common  factor  of  A  and  B. 

From  the  nature  of  subtraction,  the  minuend  is  equal  to  the  sub- 
trahend and  remainder.     Hence,  A  =  mB  -\-  D,  B  zz  n D  +  E,  and 


Process. 

B)A  (m 
mB 
D)B{n 
nD 


E)D(r 
rE 


HIGHEST  COMMON  FACTOR.  151 

D  =  r  E.  Since  the  division  has  terminated,  E  is  a  divisor  of  D. 
E  is  also  a  divisor  of  n  Z)  (Principle  1)  and  of  n  D  -{■  E^  or  B 
(Principle  2).  Hence,  £  is  a  divisor  oi  mB  (Principle  1),  and  of 
mB  +  D,  or  A  (Principle  2).  Therefore,  £  is  a  common  factor 
of  A  and  B. 

We  must  now  show  that  E  is  the  highest  common  factor. 

Every  divisor  of  A  and  B  is  also  a  divisor  of  m  B  (Principle  1), 
and  of  A  —mBy  or  2)  (Principle  2).  Therefore,  every  divisor  of 
A  and  i5  is  a  divisor  of  nD  (Principle  1),  and  of  B  —  nD,  or  E 
(Principle  2).  But  no  divisor  of  E  can  be  of  higher  dej^ree  than  E 
itself.     Therefore,  E  is  the  highest  common  factor  of  A  and  B. 

66.  Let  i4,  B,  />,  E,  etc.  represent  any  polynomials.  Let  m 
represent  the  H.  C.  F.  of  A  and  B,  n  the  H.  C.  F.  of  m  and  Z),  and  p 
the  H.  C.  F.  of  n  and  E^  etc.  Evidently  m  is  the  product  of  all  the 
factors  common  to  A  and  B  ;  also,  n  is  the  product  of  all  the  factors 
common  to  m  and  Z),  and  /)  is  the  product  of  all  the  factors  common 
to  n  and  £,  or />  is  the  product  of  all  the  factors  common  to  A,  By  D, 
and  £,  etc.,  which  is  their  H.C.F.     Hence,  in  general, 

To  Find  the  H.  C.  F.  of  Several  Polynomials.  Find  the 
H.  C.  F.  of  two  of  them;  then  of  this  result  and  one  of  the  remaining 
polynomials;  and  so  on.  The  last  result  found  ^vill  be  the  H.C.F. 
of  the  given  polynomials. 

Exercise  58. 
Find  theH.C.F.  of : 

4-  4  y*,  and  2  a;^  +  2  i/. 

2.  ai^  +  a^-Sa^-^Jx-9,  2-^  x -\- 2^ -{- s^-\- 2x^  +  2x^, 
and  3  +  3a:2_^^4.^^4^ 

3.  m"  a:3  4.  2  m"  ar2  +  m"  J?  +  2  m",  2  x'^  +  6  2^  +  4:  s^, 
Za^-h93^+9x-\-6,  '6x^-  l23^-Sx^-6x,  and  Sx^ 
-\-2-\-5x-\-S2^. 


152  ELEMENTS   OF  ALGEBRA. 

4.  2x^-5  x+  6-  3x\  3  2;2  +  2  a;3  -  8  :i:  -  12,    and 

5.  Sa^""  -  33  a;2"  +  96  a;"  -  84,  68  x^"  -  92  x^""  -  24a;» 
+  32  x!^\  a;3"  +  11  ^"-6-6  :>j2«,  50  a;"  +  20  r?;3«  -  60  r2;2H 

-  20,    5  x^"  -  10  a^"  +  7  a;"  -  14,    and    3  ^«"  -  35  a:^" 
+  162  2:^"  -  372  a;3«  +  494  a;2»  -  192  x\ 

6.  9  a:2«  +  4  X"  +  2  2:3"  _  15^    48  ^n  +  30  _  343  ^" 

-  24  x^"",    8  a;2«  +  4  x^''  +  3  a;"  +  20,  and  2  x^""  +  12  a:^" 

-  94  a;"-  60. 

7.  o  3^  —  2  X y^  —  5  x^ I/,  5  xy^—  6  y^  —  3  x^y"^  —  x^y 
+  x\  9  a^  -  8  x^ y  -  2i)  X7/,  S  xy^  -  7  x^y  -  2  y^  +  3x^ 
10  y^  —  x^y^  —  5  x^y  +  S  x!^  —  7  xy^,  and  x^  —  x^y  —  x^y^ 

-  x]^—  2y^. 

Miscellaneous  Exercise  59. 

Note.  When  possible  the  student  should  separate  the  given  expressions 
into  their  factors  by  inspection. 

rindtheH.C.r.  of: 

1.  7^  —  xy^  and  x^  +  x'^y  ■\-  xy  +  y\ 

2.  x^  —  ?/2,  {x  —  ?/)2,  of  —  x'^y,  and  2x'^  —  2xy. 

3.  2  x^  —  X  —  1,  X  y  —  y,  x'^  y  —  X  y,  and  3  x^  —  x  —  2. 

4.  rc6  -  6  X  +  5,  2a^+  b-8x  +  x\  x^  +  x^  -  11  x 
+  9,   and   42  2^2  +  30  -  72  x. 

5.  a;2_i8a;+45,  22:2-7a:+3,  2^2-9,  and  33p-7x-6. 

6.  6a;4"- 3:r3n_^2n_^n_;^  ^^^  3a^^  -  3a^'' -  2x^'' 

-  cr"  -  1. 


HIGHEST   COMMON    FACTOR.  153 

7.  a^  -  //3^  x^  +  ^V  4-  2/^  a:^  -  //,  x^  +  x//  +  i/, 
a^  +  sH^y  —  x-^  —  i/y  and  r^//  4-  ar^y^  +  .^3^. 

8.  2ar»+2rt  +  4aa;,  x^+23^+2x+l,  7b+Ubx 
+  7  6^-3+  14  6r2,  3  :r2  -  (3  m  +  n  -  3) a:  -  3  m  -  n, 
a^-_  2  2^2  4-  1,  and  2  2:2+  (2;?  +  ^  +  2) a:  4-  2p  +  q. 

9.  a:*  -  27  &8  ic,  (2:2  _  3  5  ^)2^  a  a^  -  a  b  x^  -  6  a  b^  x, 
and  &2:*-4  62a:3  4.  353^, 

10.  4  2:4'»-2  2:3«_^  3a.»_9  ^nd  2  a:*"  +  2:2'^  -  2  2:3» 
+  3  2:*  -  6. 

11.  (a  +  6)  (a  -  ^/),  («  +  5)  (6  -  a),  and  (ft  +  «)^(^^  -  ^)^. 

12.  2  63  _  10  a  &2  _^  8  a2  6,  4  a2  _  5  «  J  4.  ^,2^  ^4  _  ^,4^ 
9a*-3a68+3a252_9a8j^  and  3  a^- 3  a26  +  aJ2_  j3 

13.  3  2:3'*-3m2.'2'»  +  2m2a:''-2m8  and  3a:3"+ 2m2a:* 
+  8  m3+  12  ma:3« 

14.  (ni  —  n)  {x — y),  (m  ^n)(j/  —  x),  and  (n  —  m)  {x—y). 

15.  9  2r»+ 3  2:3+  12a:+ 20  +  2:^  3  2:2^2  +  ^^  _^2  2r» 
+  12y2+4y  +  8,  G2:2^_a;6_|.62^^32,4.24anda^»2^ 
+  3  2:2y  +  4  2:2  +  4  y2  +  12  y  +  16. 

16.  a3-+3a2'»6«  +  3a'»62m_^^«^  Sa^^+oJ^'",  4a2"fe2'» 
+  12  a"fe3m  +  8  ft*"",  and  a2--  62«. 

17.  2;*  —  ?n.  2:3  _|_  (^  _  1)  a;2  4.  ^  3.  _  ^  ^j^^-l  a:*  __  ^  ^ 
+  (w  —  1)  2:2  ^  ,j  2;  _  ^ 

18.  3n2ar»+  12?;i27i2+  3n3^-  15mn^x+  \2m^nx 
—  Ibmnx^  and  2  m  71 2:3  4.  g  ^^3  1^2  _  2  n^  a:3  4.  g  ^3  ^  2; 
+  2  m  n^ x^  —  ^  m? n^ X  —  2  noi^  —  ^  m^ nx^. 


154  ELEMENTS   OF  ALGEBRA. 

19.  x'^  —  ma^  —  mi?  x^  —  iii^  x  —  2  m^,  a;^  —  6  ni^  +  mx, 
x"^  —  2  m^  —  m  x,  3  3^  —  7  m  x"^  +  S  m^  x  —  2  m^,  and  x^ 

—  8  7n'^  +  2  7?i  X. 

20.  12x^i/-24.3^2/+ 12x^7/,  {x^y-xf'f,  xy{x^-ff, 
and  ^7^y^-2^x^if^  2^x^i/-^fx. 

21.  a^-  2o?h  -  aV^+  2h^,  a^  +  a^h  -  ah^  -  h\ 
«3  _  3  a  62  +  2  63,  a^  -h^,  2>ac-3hc  +  2ah-  2V\ 
a^-b\   and   2h^  +  a  c  -  he  -  2  ab. 

22.  a2  _  (^  ^_  c)2^  (^  4.  ^)2  _  j2^  c2  -  {a  +  hf,  and  a^ 
+  2a6  +  62+  2  6c  +  6-2  +  2ac. 

23.  a^e^  +  a^s^-Se,  ^4'^+ 5  2:3«_|_  g  ^n^  :i:6«_4  2,3"_96^ 
^3«_|.  32:2«  +  3a:"+2,  a:^'^  -  9  2:2"  +  20,  ^nd  3a:3n^3^2. 
+  5  2f^  +  2. 

24.  ^  -  2  2^2  +  3  ^  _  6  an(i  ;z;4  -  a?3  -  ^  -  2  oj. 

25.  4  ^  2/^  ~  2  2/3  +  6  ^2  ^  and  4  ^2  ^  ^  ^  x^  —  A.xy^. 

26.  35^4-47^2_|_i3^+X  and  42^4+41;2^3_9^2_9^_l 

27.  m7z,3+2??z7i2+7'/i'/i+2m  and  3?i^— 12 ?^3_ 37^2^5^^ 

28.  2m22/5+166m22/2-96m2  2/  +  108m2  and  ^mTv^f 

—  144  m  1^ y^—l^m 7? ?/2  —  108  m  n^. 

29.  2^4_6^.3^3^2_3^,+l  and  ^7_3^6+^_4^2+i2;r-4. 

30.  4a;H322;3+36^2^8^  and  8^6_24^4+24a;2-8. 

31.  a;^"— 82/3"»aj2«__2;"2/'»  +  2/'"  and  ^2«__4aj«^+42/2m 


LOWEST  COMMON  MULTIPLE.  155 


CHAPTER   XIII. 
LOWEST  COMMON   MULTIPLE. 

67.  A  Multiple  of  a  uumber  coutains  all  the  factors  of 
the  j^aven  number  with  higJiest  powers. 

Thus,  since  24  =  2»  X  3,  2«  X  3  is  a  multiple  of  24. 

A  Common  Multiple  of  two  or  more  numbers  contains  all 
the  factors  of  the  given  numbers  with  highest  powers. 

Thus,  since  12  =  2^  X  3  and  9  =  3^^,  2^  X  3^  is  a  common  multi- 
ple of  12  and  0. 

The  Lowest  Common  Multiple  (L.  C.  M.)  of  two  or  more 
algebraic  expressions  is  the  expression  of  lowest  degree 
which  can  be  exactly  divided  by  each  of  them. 

Thus,  6  a*x»y«  is  the  L.  C.  M.  of  6  a*,  x  y\  a:»,  and  a«y«. 

Example  1.  Find  the  L.  C.  M.  of  42 a^x y*,  56  a  x*y^,  63  a«x*i/«, 
and  21  a*  x^y. 

BolutioiL     Separating  the  expressions  into  their  factors,  we  have 

42  a»a?  y*  =  2  X  3    X  7  X  a*^  X  x    X  y*, 
56  a  x*y^  =  2*  X  7  X  a    X  x*  X  y^, 

63  a»x«y«  =  3«  X  7  X  a»  X  x«  X  y», 

21  a*x»y  =         3    X  7  X  a*  X  x«  X  y. 

2*  X  3*  X  7  is  the  least  common  multiple  of  the  coefficierts  42,  66, 
63,  and  21  ;  a*  is  the  lowest  power  of  a  that  can  be  evenly  divided  by 
ich  of  the  factors  a*,  a,  a\  a*  ;  x*  is  the  lowest  power  of  x  that  can 
Ihj  evenly  divided  by  each  of  the  factors  x,  x*,  x^,  x^  ;  y*  is  the  lowest 
power  of  y  that  can  be  evenly  divided  by  each  of  the  factors  y*,  y^,  y*, 
y.     Hence,  the  L.  C. M.  =  2»  X  3^  X  7  X  a»  X  x«  X  y»  =  504a«x*y«, 


156  ELEMENTS   OF  ALGEBRA. 

Example  2.  Find  the  L.O.'M..  oi  6x^-2 x,  9x^-3 x,  t5{z^+x y), 
8  (x  z/  +  y'^y\  and  \2  a^  x^  y^. 

Solution.  Separating  the  given  expressions  into  their  factors,  we 
have 

12  a^-x^y'  =  22  X  3  X  a2  X  a:3  X  y^ 
8(^2^  +  3/2)2^23  X  2/2  X  (a:  +  y)2, 

6(a;2  +  ar2/)  =  2  X  3  X  a;  x  (a;  +  2/), 

6a;2-2a;=2  x  a;  X  (3  a;-  1), 

9a;2-3a:=  3  X  a;  X  (3  a;-  1). 

23  X  3  is  the  least  common  multiple  of  the  coefficients ;  aP-  is  the 
lowest  power  of  a  that  can  be  evenly  divided  by  a^  ;  ofi  is  the  lowest 
power  of  X  that  can  be  evenly  divided  by  each  of  the  factors  x\  x,  x,  x. 
Similarly  2/,  (a?  +  2/),  and  (3  a:  -  1),  each  affected  with  its  highest  ex- 
ponent, must  be  used  as  multipliers. 

Therefore,  the  L.  C.  M.  =  2^  X  3  X  a2  x  a;^  x  2/^  X  (x+yf  X  (3 a;- 1) 
=  24  a^x^y^  (x  +  y)"  (3  x  -  1). 

Example  3.  Find  the  L.C.M.  of  4  aa;2  2/2  +  n  aa;2/2  -  3a  t/^ 
a;8  +  6  a;2  +  9  a;,  3  x^  y^  +  7x'^y^-6xy%  and  24  aa;2- 22aa:+ 4  a. 

Process. 

4aa;22/2+ llaa;2/2-3a2/2=     a     X  y^(x  +  3)  X  i^x-1), 
a;3  +  6a;2  +  9  a;  =       xX      (x  +  3)2, 
3a:32/3  +  7a;22/3-6a;.?/3=       x       y^(x  +  3)X  (3a;-2), 

24aa;2- 22aa;  + 4a  =  2a  X  (4a;-l)  (3a;-2). 

.-.  the  L.C.M.  =  2aa;iy3(a;  +  3)2  (4  a;-  1)  (3  a;- 2).  Hence,  in 
general, 

To  Find  the  L.  C.  M.  of  Two  or  more  Expressions  that  can 
be  Factored  by  Inspection.  Separate  the  expressions  into  their 
factors.  Take  the  product  of  the  factors  affecting  each  with  its 
highest  exponent. 

Note.  The  L.  C.  M.  of  two  or  more  prime  expressions  is  their  product. 
Thus,  the  L.C.M.   of 

(i^  +  ab  +  b^,  aS  +  b%  and  a^  +  b^  is  (a^  +  ab  +  b^)  (a^  +  62)  (^s  +  J8). 


LOWEST  COMMON  MULTIPLE.  157 

Exercise  60. 
Find  the  L  CM.  of: 

1.  4&3^i/,  ^^aT^f,  and  63r/^2«. 

2.  24:111 71^3^,  Z(Sm^n^2^,  and  4871828. 

3.  Ua^l^c^,  9aHc2,  and  ^(Sah^d^. 

4.  12m*n2  2/3,  l^mnf,  and  24m^?i3. 

5.  12aa;8y*,  a:'"^  —  ?/-,  2;-—  2a:y  +  ?/,  and  rr^  +  2a;^  ^-  2/2. 

6.  m^  (a;^  —  ^),  71^  (^x  —  y),  and  a:"*  —  y^. 

7.  2a:(a;-  y),  -ixyix^  --?/),  and  62:3/2(2.4.  y)^ 

8.  2:2  +  a:  -  20,  2^^  -  10  a:  +  24,  and  2:2  _  2:  _  30. 

9.  2^2+22:,  2:2  + 4a: +  4,  2:243^42,  and  a^ -\- o x -^^  e,. 

10.  2:*  +  a2  2:2  +  a*  and  2-'*  —  a  a:^  —  ft^  a;  +  a*. 

11.  a:2  _  3  2;  -  28,  2^2  +  2:  -  12,  and  a^  -  10  a:  +  21. 

12.  15  (2:2^  -xy%  21(2^-  ar/),  and  35  {xy"^  +  f). 

13.  ar2  -  1,  ar^  +  1,  and  a^  -  i. 

14.  '6x^-\-  llx-^  0,  32:2  4.  3^,  _j.  4^  g^^j  2r^  +  52-  +  6. 

15.  2^+{a-^b)x-\-aby  a^-\-(a-\-c)x+ac,  and  a^-{-(b-\-c)x-{-bc. 

16.  mx  —  my  —  nx  +  ny,  (x  —  y)^,  and  Zm^n—?nnn^. 

17.  a:2  4.  (rt  4  2,)  2,  4  a  5  and  ar2  +  (a  -  i)  a:  -  a  6. 

18.  ar2  -  1,  a^2  4  1,  2:4  4  1    ^nd  a:^  +  1. 

19.  a:^  +  ar*y  +  a;?/2  -f  ?/3^  ^i  —  x'^y  -{-  xy^  —  if,  and  ^ 
-\-  x^y  -xf-f. 


158  ELEMENTS  OF  ALGEBRA. 

20.  6  aa^+7  a^x^-S  a^x,    3  a^  x^  +  Ua^x-^  a\ 
and  6x^  +  39ax  +  45  a^. 

21.  x^  +  5  X  +  4:,  x^  +  2  X  -  8,  and  a:^  +  7  ic  +  12. 

22.*   12  x^  -  23  a:^  +  10  2/^,  4a:2  _  9^^  _j_  5^2^  ^nd  30^2 
-—  5xy  +  2y^. 

23.  a'-^-4&2,   a3-2a2Z^+4a&2_8j3^  and  a^+2a^b 
+  4  a  62  +  8  53 

24.  a  m  +  <z  '/I  +  &  /?i  +  &  rt  and  ax  +  ay  +  hx-^hy. 

25.  8a;2_38a;2/+352/^  4a:2_:i,z/_53/2  and  2x^-5xy-7y^ 
26. "  2^2  +  ?/2,  x^  —  n:2  2/2  +  2/^  and  2:^  +  y^. 

27.  60  a;4  +  5  a;3  _  5  ^2^    60  n;2  ?/  +  32  a;  2/  +  4  y,    and 
40  a:^  2/  ~  2  a:2  2/  —  2  a;?/. 

28.  {a  +  6)2  -  (c  +  ^)2,  (a  +  c)2  -  (&  +  ^)2,  and  (a  +  df 
-  {b  •+  c)2. 

29.  2^2  -{_  ^  ^  _|_  ^2^  2^  _  ^  y  _l_  y2^  and  a:^  +  a;2?/2  +  2/*. 

30.  3  a^*  +  26  a;3  +  35  a;2^  6  a:2  +  38  a:  -  28,    and   27  x^ 
+  21x^-30x. 

31.  12  a:2n  _^  3  ^n  _  ^g,    -^^  ar^«  +  30  a;2"  +  12  x\    and 
32^2n_4Q^„_  28 

32.  «  (??/  —  n),  b  (n  —  m),  and  —  c{m  —  n). 

33.  (a  -b)(b-  c),  (b  -a)(b-  c),  and  (b  -a)(c-  b). 

34.  a(&  — a:)(a;— ^),  b{c—x){x—a),  and  c(a  — a:)  (a:— 6). 

35.  a:*^  -  2  aj2n  _,.  1  ^nd  a:*"  +  4  x^''  +  6  a^"  +  4  a:"  +  1. 
Result.     a;«"  +  2  ar^"  -  a;^"  -  4  rr^"  -  x^"  +  2  a;"  +  1. 


LOWEST  COMMON  MULTIPLE.  159 

68.  If  the  expressious  cannot  be  factored  by  inspection,  find  their 
H.C.  F.,  then  proceed  as  before.     Thus, 

Example  L  Find  the  L.  C.  M.  of  2  z<  +  a:«  -  20  x"  -  7  x  +  24 
and  2  X*  +  3  z»  -  13  x2  -  7  X  -h  15. 

Solution.  The  H.C.F.  of  the  expressions  (Art.  64)  is  x^-\-2x-3. 
Dividing  each  expression  (for  the  other  factor)  by  x*  +  2  x  —  3,  we 
have  2  X*  -  3  X  —  8  and  2  x*  —  x  -  5.     Hence, 

2  x«  +  x»  -  20  x2  -  7  X  -h  24  =  (x2  -h  2  X  -  3)  (2  x«  -  3  X  -  8), 
2  X*  +  3  x«  -  13  x2  -  7  X  -h  15  =  (x*  -f  2  X  -  3)  (2  x'^  -     x  -  5). 

.-.  the  L.C.  M.  =  (x2  -H  2  X  -  3)  (2  x2  -  3x  -  8)  (2  x2  -  X  -  5). 

Example  2.  Find  the  L.C.M.  of  x«  -  Sx^  -f-  19z  -h  12,  x»-  6x2 
+  11  X  -  6,  and  x«  -  9  x2  -f  26  X  -  24. 

Solution.  The  H.C.F.  of  the  expressions  (Art.  66)  is  x  -  3. 
Dividing  each  of  the  expressions  by  x  — 3,  and  factoring  the  quotients, 
we  have 

x»-8x«+19x-12  =  (x-3)(x2-5x  +  4)  =  (x -3)  (x-1)  (x-4), 
x»-6xHnx-  6=  (x-3)(x2-3xf2)  =  (x-3)  (x-1)  (x-2), 
x«-9x2-|-26x-24=  (x-3)(x2-6x-f8)  =  (x-3)  (x-2)(x-4). 
Therefore,  the  L.C.M.  =  (x-3)(x-l)(x-2)(x-4) 

=  x*-10x»+35x2-50x  +  24.     Hence, 

To  Find  the  L  C  M.  of  Two  or  more  Polynomials  that  can- 
not readily  be  Factored  by  Inspection.  Find  the  H.C.F.  of  the 
^^iven  polynomial!*,  and  divide  each  polynomial  by  it.  Then  find  the 
L.  C.  M.  of  their  quotients,  and  multiply  it  by  the  H.  C.  F. 

Exercise  61. 
Find  the  L.  CM.  of: 

3.   x^-{-2.x-3,  :r?-\-Sx^-r:-S,  and  a:^  +  4 r^  +  a: - 6. 


160  ELEMENTS  OF  ALGEBRA. 

4.  ic*  —  m aj3  —  ni^ x^  —  m^ x  —  2m^  and  3  a;^  —  7 m  cc^ 
+  3  m^  5::  —  2  m^. 

+  38^2/3+  16^2/*-  10  2/^- 

6.  a?3  -  9  :c2  +  26  ^  -  24,  a;3  -  10  a:2  +  31  2;  -  30,  aud 
aj3- 112^2+  38:^-40. 

7.  ^-4 -2)3- 4 a;2+  16  2^-24,  2^3_  5^2  ^_  8^-4,  and 

a:2  +  2  2:  -  8. 

8.  2)3  _^  ^.2  _  10  a:  4-  8,  ic2  +  2  a:  -  8,  a;2  -  3  a;  +  2,  and 
ic2-  1. 

9.  6  2;3  +  15  0^2  _  6  ^  4.  9  and  9  a;3  4.  G  2,2_  51  ^  4.  36^ 

10.  2  ar^  -  8  a;4  +  12  a;3  -  8  2:2  +  2  a:,  3  a;5  -  6  2^  +  3  re, 
aud  a::3_3^2_|_3^_l 

11.  a:*  +  5  2^3  +  5  rc2  -  5  2:  -  6,  a:^  +  6  a:2  +  11  a;  +  6, 
and  a:^  +  4  2^2  +  2^  —  6. 

12.  2  2)3  +  7  a;2  +  8  X  +  3,    2  x^  -  2^2  -  4  2;  +  3,    2  a:^ 

+  3  2:4  +  2  x^  +  3  ^,2  4_  2  a:  +  3,  and  a?*  4-  2^2  +  1. 

69.  To  Prove  the  Method  for  finding  the  L.C.M.  of  any 
Two  or  more  Algebraic  Expressions.  Let  A,  B,  D,  E,  etc. 
represent  the  expressions,  F  represent  their  H.  C.  F.,  and  M  represent 
their  L.  C.  M.  Also,  let  a,  b,  d,  e,  etc.  represent  the  respective  quo- 
tients when  Af  B,  D,  etc.  are  divided  by  F.     Then, 

A  =aF,  B  =  bF,  D  ^  d  F,  E  =  e  F,  etc.  (1) 

F  is  the  product  of  all  the  factors  common  to  A,  B,  Z>,  etc.  The 
quotients  a,  b,  d,  e,  etc.  have  no  common  factor.  Hence,  their 
L.C.  M.  is  a  6  rf  .  . . ,  etc.  and  the  L.  C.  M.  of  aF,  bF,dF,  etc.,  or 
their  equals  A,  B,  D,  etc.,  is  ah  d  ...  F.  Therefore,  M  —  abde  F, 
etc. 


LOWEST  COMMON  MULTIPLE.  161 

70.  Let  A,  B,  D,  Ej  etc.  represent  any  polynomials.  Let  A' 
represent  the  L.C.  M.  of  A  and  B,  P  the  L.C.  M.  of  N  and  D,  and 
R  the  L.  C.  M.  of  P  and  Ej  etc.  Evidently  R  is  the  expression  of 
lowest  degree  which  can  be  divided  by  P  and  E  exactly ;  also,  P  is 
the  expression  of  lowest  degree  which  is  exactly  divisible  by  A^  and 
Z>,  and  N  is  the  expression  of  lowest  degree  which  is  exactly  divisible 
by  A  and  B.  Therefore,  R  is  the  expression  of  lowest  degree  which 
is  exactly  divisible  by  ^4,  ^,  2>,  and  E,  etc    Hence, 

To  Find  the  L.G.M.  of  Several  Polynomials.  Find  the 
L.  C.  M.  of  two  of  theni;  then  of  this  result  and  one  of  the  remaining 
expressions;  and  so  on. 


71.  Let  A  and  B  represent  any  two  expressions.  Let  F  repre- 
sent their  H.C. F.,  and  M  represent  their  L.C.  M.  Also,  let  a  and  b 
be  the  respective  quotients  when  A  and  B  are  divided  by  F.  Then 
A  =  aF,  B  =  bF,  and  M  =  ab  F.  Multiplying  the  first  equations 
together  (Axiom  3,  Art.  47),  we  have  AxB  =  aFXbF=FXabF. 
Therefore,  substituting  for  abF  its  value  M,  A  B  -  F M.  Hence, 
in  general, 

The  Product  of  any  Two  Expressions  is  Equal  to  the 
Product  of  their  EOF.  and  L.C.M. 


Miscellaneous  Exercise  62. 

Find  the  L. CM.  of: 

aZ^a^h-ah^-h^  and  a^  -  2  aH  -  a  6^  +  2h\ 

2.  a:4«  _  10  2:2-  4.  9^  ^n  4. 10  r^"  4-  20  ar**  -  10  a:"-  21, 
and  a;*-  +  4  a;8"  -  22  ar»"  -  4  af  4-  21. 

3.  2:3"-4ar2"3r+ Oa:"^^"'- lOyS"  and  a:«*-f2a;2nym 

11 


162  ELEMENTS  OF  ALGEBRA. 

4.  s^  -^  Sx'^  +  x^  +  3x^  +  x  +  S,  2a^+6x^^2x-6, 
r"  +  2x^  +  a^+  2x^  +  x+2,  and  2a^+3x^+2a^  +  3x'^ 
+  2;:c+  3. 

5.  X  y  —  b  X,  X  y  —  a  y,  i/^  ^  3  h  y  -\-  2  h^,  x  y  ~  2  h^, 
xy  —  2hx  —  ay  +  2  ah,  and  xy  —  hx  —  ay  -\-  ah. 

6.  a^"  +   ^^4"  h"^  +   a3»  52m  _j_  ^2n  53m  _^  ^n  54m  ^  j5m^    ^nd 

7.  .:c2'*  -  4  a'-^"*,  .:c3''  +  2  a"'  *2«  +  4  a^'"^;"  +  8  a^^     and 

^3«  _  2  ^'«  ^2»  ^_  4  ^2m  ^«  _  3  ^3m 

8.  27^""  +  {2a-3  &)^2n  _  (2  52  +  3a&)a;"  +  3  ^)3  and 
2a;2"- (3  6-2  c)^?'^- 3  6c. 

9.  ^- 2:r2_^  4^_8^  ^3+2^2_4^_8^  a^-3a^ 
-4:X  +  12,  and  ^^  -  3  a;*  -  20  a;^  +  60  aj2  +  64  a;  -  192. 

10.  x^''-{a-h)x^-ah,  x'^''-(h-c)x''-hc,  a^^'-x^^'h^ 

-  a;6»62  +  58^  and  a)2»  -  (c  -  a)x''  -  ac. 

Find  the  H.  C.  F.  and  L.  C.  M.  of : 

11.  3  o[^  —  7  x^y  +  5xy^  —  y^,    x^ y  +  3  xy^  —  3  0^—1^, 

and  3  a^  +  5  x^y  +  xy^  —  y^. 

12.  6^'5  4.i52,4^_4^2^_;l^Q^2^2^y  and9^y-27a^22^ 

—  6  a2a;?/  +  18  <x^z/. 

13.  6^3«  +  ^2n_5^n_2  and  6  2^"  +  5  2;2~-3.2J*-2. 

14.  a^  -ah  +  h^  a^  +  ah  +  h\  a^  +  a^ h'^  +  6^  a^  +  j3^ 
a^  —  6^,  and  (c^  —  h^f. 

15.  2rc2^(6^_io6)a3-30a6  and  3  a;2- (9  a  +  15  6)  a; 
+  45  a  6. 


LOWEST  COMMON   MULTIPLE.  163 

16.  a:3n_  92«J- ^  26 X-'*  -  24  and  a:3»_  122:2-+  47 ^•"-CO. 

17.  (a.-2  +  62)c  +  (62  +  c2)x  and  {x^  -  W)  c  +  {h^  -  c^)  x. 

18.  (2  2^2  _  3  ,,i2j  y  4.  (2  m2  -  3  y2)  a;  and  (2  m2  4.  3^2)^ 
+  (2x2+3m2)y. 

19.  2:3«^  2a'"a:2n_^  a2'"a:"  +  2  a^^  and  a:^"  _  2  a'"aj2« 
+  a2m^„_2a3'". 

20.  a;3 +3a^y  +  3a;^^/J  + 2/,    oi^  -  x^y  ^  xy^  -  2f, 
sc^  -\-  xy  -\-  y^y  and  «*  +  2^2  ^2  _^  ^ 

21.  20^^x^-h  25a:<  +  5a:8~2:~l,  and  25 a;*-10 0:24.1, 

22.  2:8* —  yS*      a^s-y-  — i/*",     y^(af  —  f)^,    and    a:2« 
-  2  a;*y"  +  2^'". 

Find  the  L.C.M.  of: 

23.  a:*-7a:8_7-2^_^43a.4.42  and  ar*-9a:8+9a:3 
H-  41  a;  -  42. 

24.  ir3+4a:a+6a;+9,  «:«+«»- 2 a; +12,  and  a:2_^_i2. 

25.  4a:«-42:*~29a^»-21  and  4a:«+ 24a;* +  41 2:24.21. 

26.  2a:*-ll2:8+32r24.i0a;  and  32:*-142:«-6a;24.5a., 

27.  2:3  _  6  2-2  4.  11  a;  _  e,     a;3  _  a:2  _  14  a;  +  24,     and 
2:3  +  .r2  -  17  2:  +  15, 

28.  32:^  +  52:84.525^52.^2  and  32^-2:8+2:2_2._2. 

29.  9a:*+18a:3_.^_92^4.4  and  6a:*+17a:«-10a:+8. 

30.  2  m^  +  m2  -  m  +  3  and  2  m8  +  5  m2  ~  w  -  6. 


164  ELEMENTS  OF  ALGEBRA. 


CHAPTER   XIV. 

ALGEBRAIC  FRACTIONS. 

72.  The  expression  (a  +  &)  -f  (m  +  n)  may  be  written  — 77- . 
It  is  read  the  same  in  each  case. 

The  second  form  is  called  a  Fraction  ;  the  dividend  is  the 
Numerator,  the  divisor  is  the  Denominator,  and  the  two  taken 
together  are  called  the  Terms  of  the  fraction,  a,  b,  m,  and  n  may- 
represent  any  numbers  whatever.     Hence, 

An  Algebraic  Fraction  is  an  indicated  operation  in  divi- 
sion. 

A  Mixed  Expression  is  one  composed  of  entire  and  frac- 
tional parts ;  as,  n 

m-\ 

a 

Note.  The  dividing  line  has  the  force  of  a  symbol  of  aggregation,  and  the 
sign  before  it  is  the  sign  of  the  fraction  and  belongs  to  its  algebraic  value. 


73.  Multiplying  or  dividing  the  divisor  and  the  dividend  by  the 
same  number  does  not  change  the  quotient.  For,  if  we  multiply  the 
dividend  by  any  number,  as  m,  the  quotient  will  be  increased  m 
times  ;  if  we  multiply  the  divisor  by  m,  the  quotient  will  be  dimin- 
ished as  many  times.  A  similar  method  of  reasoning  may  be  applied 
to  the  dividend  and  divisor. 

A  fraction  is  in  its  lowest  terms  when  the  numerator  and 
denominator  have  no  common  factor. 

7a^hc 
Example  1.     Reduce  ^^^   3,3  to  its  lowest  terms. 

^o  0/    0  C 


ALGEBRAIC  FRACTIONS.  165 

Solution.     The  H.C.F.  of  the  numerator  aud  the  denominator 

7a^bc  X  1 
is  la^bc.    Factoring,  we  have  7^2^^  X4ac'    ^^J^^^"^8  ^^^  H.C. F., 

we  have Since  the  terms  are  prime  to  each  other  the  fraction 

4ac 

is  in  its  lowest  form. 

6  a*  +  ax- 15  x^ 

Example  2.     Reduce  ,r:    2  .   ^u^^ — TTZz  to  its  lowest  terms. 
15  a*  +  lis  ax  —  15  x^ 

6a^-\-  ax  -  15  x'     _  (3  a  +  5  x)  (2  a  -  3 a;) 
^^°*^®**'     15a2+  16aa:-15x2~  (3a4-5a;)(5a-3x) 

_  2a  -Sx 
-  5a-3x' 

Explanation.   Dividing  the  terms  of  the  fraction  by  their  H.  C.  F., 

we  have This  result  is  in  its  lowest  terms,  since  the 

5a  —  3x 

numerator  and  denominator  have  no  common  factor. 

x*-j-x^y-\-xy*  —  y* 

Example  3.     Reduce  -7 — ^ 5 -.  to  its  lowest  terms. 

XT  —  x*y  —  xy*  —  y* 

x^  +  x^y+xy'-y*  _  (x*  -  y*)  +  (x*y  f  xy*) 
Process.    ^^^y_^^_yA  -  (^  _  y4)  _  (^^  ^  ^^.^ 

^  (x^  +  y«)  (x«  -y^)  +  xy  (x«  +  y«) 
l^  +  y«)  (a:«  -  ya)  -  xy  (x«  +  y^^ 

^(x«  +  y«)[x«-y«  +  xy] 
(x2  +  y«)[x2-y2_^yj 

x^  +  xy-y^ 


When  the  factors  of  the  numerator  and  denominator  cannot  be 
readily  found  by  inspection,  their  H.  C.  F.  may  be  found  by  the 
method  of  Art.  64,  and  the  fraction  then  reduced  to  its  lowest  terms. 
Thus, 

.      T>   ,         4a»+  12a«6 -afta- 15!»»         .      , 

Example  4.     Reduce  ^    .  ,   ..^    «. -. — i» — Trut  to  its  lowest 

6  a"  +  13  a*o  —  4  ao*  —  15  6* 
terras. 


166  ELEMENTS  OF  ALGEBRA. 

Solution.  6a^+r3a^b-4ab^-  1568)  4a^+l2aH~    ab^-l5b^(^2 
3  times  the  numerator,  1 2 a^  +36 a^lSab^  —45 b^ 

2  times  the  denominator,  I2a^-j-26a^b-8ab^-30b^ 

First  remainder,  lOa^b  +  bab^—lbb^ 

=  5b(2a^+ab-Sb^). 

2a2  +  a&-362)6a8+  13a26-4a62_  i568(3a  +  56 
6as+    3a2&-9a62 


10  a^b  +  5  ab-^-  15  b^ 
10  a^b-{~  5  ab^-  15  68 


.'.  the  H.  C.F.  of  the  numerator  and  denominator  is  2a2+a6— 36^. 
Dividing  each  term  of  the  fraction  by  2  a^  +  a  6  —  3  6^^  we  have 
4a8+ 12a26-a62_i5  58       2a +  56 


6a8+ 13a26-4a62- 15  68      3a +  56 


Hence, 


To  Eeduce  a  Fraction  to  its  Lowest  Terms.     Divide  both 
terms  by  their  H.  C.  F. 

Exercise  63. 

Keduce  to  lowest  terms : 

75ax^y^'    ^m^a^y'^^    4:X^  +  6  xy' 

72  m^n^x^\    mn^{a^-y^f  ^    2^  +  Sx+  1 
24:'m'nix'''    w?n{a^  —  j^)'      x^  —  x  —  2 


6  m^- 11m -10  20  (a^  -  if)  af"!/^" 

6  w2- 19  m +10'    5  2:2 +5  ^2/ +  5^/2'    ^imyn+i- 

3  m^  +  23  yyi  -  36      3  772^  +  ^m^n  +  ^Trv^Ti^ 

4  m2  +  33  m  -  27  '       m*  +  m^Ti  -  2  m27i2     * 

^  in  +  Zmx       x^  —  {a  -\-  b)x  +  ab 
4  ml  —  4  mt  2:2  '    a:2  +  (c  —  ^t)  2:  —  «  c ' 


(m  +  w)2  —  a:2       ^.s  _  3  ^2y  +  3  2;  y^  _  -^yS 
m a:  +  71  a:  —  a:2'        a:^  —  x^y  —  xy^  -\-  y'^ 


ALGEBRAIC  FRACTIONS.  167 

cr^  —  (y  +  mf      ac  —  ad  —  he  -\-  hd 
x^  -\-  X y  ■{-  m X*  a^  —  b^ 

0.^4-  (g  +  b)x+  ah  27a  +  a^ 

a^Jt\a-\-c)x+  ac'    18a-6a2+2a8' 


10. 


11. 


12. 


(a  _  6)2  _  c2'    (6  4-  a;)-^  -  {a  +  c)2 

m^  —  m*  n  —  m  n*  +  71^        a  a^  —  6  ic*"*"^ 
m*  —  m^ 71  —  m^ ?i2  -|-  ??i ti^'     a^hx  —  h^a^ 


a8+3aH+ 3a62+268'    48  ar*  +  16  a:2  _  15 ' 


7*.    Example.     Reduce  :7^ :r— ^ —  to  a  mixed 

expression. 

ProcesB. 

2x  -  3y  ) 4x«  -  16x2y  +  29a;i/2 -  22y« (  2z2  -  sxy -f  7ya  +  5-^^^^4— 

-10x2y  + 29x1/2 
-10xgy-H5xy« 

14xy^-22y« 

14xy«-21y« 

-     .y»  • 

Explanation.  Dividing  the  numerator  by  the  denominator,  we 
have  2  «*  —  6  xy  +  7y''  for  a  quotient,  and  a  remainder  of  —  y«.  As 
—  y*  is  not  exactly  divisible  by  2x  -  3y,  we  indicate  the  division  and 
atld  the  result  to  2  x*  —  5  x  y  +  7  y*.     Hence, 

To  Reduce  a  Fraction  to  an  Entire  or  Mixed  Expression. 
Divide  the  numerator  by  the  denominator. 


168  ELEMENTS  OF  ALGEBRA. 

Exercise  64. 

Eeduce  to  an  entire  or  mixed  expression 

1  +  22J     2x^-x^- 9x^+14: 
'   r^^^J^'  2x^-  x-S 

x^-2x^       6  a3  -  13  o2  +  6  a  -  6 
z.  - 


3. 


5. 


-x+  1'  3a^-2a  +  1 

x^  +  ax^-3  a^x-  S  a^     ^^3  4.  2  a:^  -  12  a:  -  13 
x-2a  '  a;2  +  ^  -  12 

x^  —  2  x^y^  +  f/     x^-\-(m  +  n+l)x  +  mn+a 
x^  +  2  xy  +  y^  '  x  +  n 

6a^-5x^+  Ix-b     a^S"  -  a:^"?/"  +  x^'y'^''  -  y^^ 
2x+l  '  ^'"  -  2/" 


75.    Every  expression  may  be  considered  as  a  fraction  whose 
denominator  is  unity.     Thus,  a  =  - ;   a^b  —  c^  = . 

2;2  _  y2  _  5 

Example.     Reduce  x  -\-  y to  fractional  form. 

^  x-y 

a:2  -  «2  _  5       X  +  y      x^  -  y^  —  t) 
Process.     x-\-y —     ~  ^  ^ 


{X- 

^V) 

X  - 
X  (rc- 

-y 

x' 

ar2- 

1 

X(x- 

-5) 

5 
X  —y  ~  X  —  y' 

Explanation.  Writing  the  entire  part  in  the  form  of  a  fraction 
whose  denominator  is  1,  and  multiplying  both  terms  of  it  by  x  —  y, 
we  have  the  third  expression.  Since  the  sum  or  difference  of  the 
quotients  of  two  or  more  expressions  divided  by  a  common  divisor, 
is  the  same  as  the  quotient  of  the  sum  or  difference  of  the  expressions 
divided  by  the  same  divisor,  we  have  the  fourth  expression.  Uniting 
like  terms,  we  have  the  result.     Hence, 


ALGEBRAIC  FRACTIONS.  169 

To  Reduce  a  Mixed  Expression  to  the  Form  of  a  Fraction. 

Multiply  the  entire  puit  liy  the  denouiinator  ;  to  the  product  annex 
the  numerator;  unite  like  terms  and  under  the  result  write  the 
denominator. 

Notes :  1.  In  the  above  example,  since  the  sign  before  the  dividing  line 
indicates  subtraction,  we  most  subtract  the  numerator,  x^  —  y^  —  5,  from 
{X  +  y)  (x  -  y). 

2.  If  the  sign  of  the  fraction  is  — ,  and  the  numerator  is  a  polynomial,  it 
will  be  found  convenient  to  enclose  it  in  a  symbol  of  aggregation  before  annex- 
ing it  to  the  product. 

Exercise  65. 
Reduce  to  fractional  forms  : 

1.  a  —  X  -\ ; — ;       -         -\-  I]    a  +  0 r- . 

a  -\-  X     m  —  n  a  —  b 

2.  „,._,„,  +  ,.-_^^;     ^^j;±_^^ -(.-,). 

,   «  —  m^n^       a   .       „   .  .    i       m*  —  1 

3.  m  n  H ;  m^  +  m^  +  ?>i  -f  1 


m  n  m  —  1 

4.  ic  +  1  4-  ;  ;    -3-^ — 3  +  1 ;  w  (a:  +  ?/)  +  — — 

7/r  — ar  x  -r  t 

2  n  (3  w2  +  ?i2)                          wi*  +  w* 
6.  m  +  w ^ — , — ^2 — ;    (w  +  ny  — r^  • 


7    a^m  _a^,^«^  y8»_ 


a^m  ^  sTir  ■\-  7/2" 


8.  .^^-,2^/+y»--^'"-'^-^"-'^°r-r-^'--^" 


170  ELEMENTS  OF  ALGEBRA. 

76.  It  may  be  sliown  by  multiplication  (Art.  22)  that : 

{+a)i+b)  =ab;       (-«)(-&)  =  ab. 

(+a)(+b){-\-c)         =abc\     {-a){-b){+c)         =abc. 
(+a){+b){+c){+d)  =  abed;  {-a){-b)(-c){-d)  =  abed,  etc.    Hence, 

In  an  indicated  product  of  any  number  of  factors,  all  the  signs  of 
any  even  number  of  factors  may  be  changed  without  changing  the  value 
of  the  product.      Thus, 

(x-y)  (:y-z)  =  (y-x)(z-y); 
(w  -x)  (x-  y)  (y~z)  =  {X  -  w)  (x  -  y)  (z  -  y),    changing    the 
signs  of  the  first  and  third  factors. 

Note.  In  order  to  multiply  a  product  containing  several  factors  by  a  given 
expression  the  student  must  be  careful  to  multiply  only  one  factor  of  that 
product  by  the  expression.    Thus,  in  order  to  multiply  both  terms  of  the 

fraction ; — '-— — ~-  by  a,  we  must  multiply  either  a-\-b  or  c  +  d  and 

m  -\-  n  or  X  -}-  y  hy  a. 

77.  It  is  often  convenient  to  change  the  order  and  the  signs  of 
the  terms  of  the  numerator  or  denominator,  or  both.     Thus, 

Change  the  order  and  the  signs  of  the  terms  of  the  numerator  and 
denominator  of  the  following  fractions  : 

b  —  a  m  —  n 

1.    -•  2. 


y  -X  '   (jc  —  h)  {x  —  m) 

Solutions  :    1.  Multiplying  both  terms  of  the  fraction  by  —1,  we 

have 

b  —  a  __  (b  —  a)  X  —I  _a  —  b 

y—x~{y~x)  X  —l~x—y' 

2.  Multiplying  the  factor  x  —  m  and  the  terms  of  the  numerator 
by  —  1,  we  have 

m  —  n  _  (m  —  n)x— 1  n  —  m 


(c  ~b){x-  m)       (c  -  b)  [(x  -  m)  X  -  1]       (c--b)(m~x)' 

Multiplying  the  factor  c  —  b  and  the  numerator  of  this  fraction  by 
—  1,  and  since  adding  a  negative  quotient  is  the  same  as  subtracting 
a  positive  quotient,  we  have 


ALGEBRAIC  FRACTIONS.  171 

n  —  m         _       (n-m)  X  —  1 +  (n  -  m) 

(c-b)(m-x)~'^i(c-b)X-l](m-x)~~  (b-c)(m-x)' 

Change  to  equivalent  fractions  having  the  letters  arranged  alpha- 
betically, and  the  first  letter  of  each  factor  in  the  numerator  and  the 
denominator,  positive : 

x  —  m  (b  —  a){c  —  a) 


3. 


{b-a)(a~c)(y-x)  "    (d  -  a)(c  -  b)(n  -  m) 


Solutions  :  3.  Multiplying  the  numerator  and  the  factor  y  —  x 
by  —  1,  we  have 

X  ~  m  _  m  —  X 

(b  -a)  (a-  c)  (y-x)~  (b  -  a)  (ti  -c)(x-y)' 

Multiplying  the  numerator  and  the  factor  b  —  a  of  this  result  by 
—  1,  we  have 


(6  -a)  (a-  c)  (x  -  y)  (a-  b)  (a  ^  c)  (x  -  y) 

4.  Multiplying  the  factors  c  —  a  and  n  —  m,  b  —  a  and  c  —  6  by 
1,  respectively,  we  have 

(6  -a)(c-  a)  (a  -  b)  (a  -  c) 


{d  -  a)  (c  -6)  (n  -  m)       (d  -  a)  (b  -c)(m-  n) 

Q;„,n„rlv  (a-h)(a-c) (a  -  b)  (a  -  c) 

oimuariy,    ^^  _  ^^^  ^^  _  ^y  ^^  -  n)  '       (a  -  d)  (b  -c)(m-n) 


(d  —  a)  (c  —  o)  (n  —  m)  (a  ^  d)  (b  —  c)  (m  —  n) 


Exercise  66. 

Change  each  of  the  following  fractions  to  four  equivalent  ones 
with  respect  to  the  signs  of  letters  : 

^     7W  —  71    _      a  —  Jj  m  VI  -\-  n  ^  a 

a  —  b^      m  +  n  —  x'    a  —  h  -\-  x'    m  —  n  -\-  a* 


172  ELEMENTS  OF  ALGEBRA. 

Change  the  following  fractions  to  equivalent  ones  having  m  and  n 
positive  in  both  terms  : 

m  —  a      a  +  m  —  X         a  +  b  —  n 
b  —  n  '     b  —  m  —  y^        a  —  b  +  m 

X  —  m  X  —  m  {a  —  m)  (b  —  m) 

y  —  n^        (y  —  m)(z  —  n)'       (c  —  m)  (x  —  n)(y  —  m) ' 

Change  the  following  fractions  to  equivalent  ones  having  the  let- 
ters of  the  terms  arranged  alphabetically  and  the  first  letter  of  each 
factor  in  the  denominator  positive  : 

.  2x-?>  —  y  3  — c  +  ft 


[m.  —  a) {2  X  —  b) (b  +  a)'        (y  —  x)  (m  —  n)  (a  —  c)' 


^  (x  —  m.)ba 

5.      r^ T7-7T- 


xy mn{c  —  b)  (b  —  a)  {c  —  a) 

{jj-x)yx 


cb a {b  —  a)  {z  —  y)  (c  —  a)  {y  —  x) (n  —  m)' 


78.   Fractions  having  a  common  denominator  are  similar. 

Thus,  ^,  — =-,  and  — ^  are  similar. 
ab    ao  ab 

2x        3  5  n^ 

Example  1.     Reduce  r — 5,  «,  and   -. — i  to  similar  fractions 

having  the  lowest  common  denominator. 

Solution.  Evidently  the  lowest  common  denominator  is  20 in^n^x^f 
the  L.  C.  M.  of  5  m^,  mn%  and  4x^.  Dividing  20  m^n^x^  by  the 
denominator  of  each  fraction,  and  multiplying  both  terms  of  each 
fraction  by  the  quotient  each  by  each,  we  have 


ALGEBRAIC  FRACTIONS.  173 

2x  2x  X  4n^x^        'Sn^x*     . 

6  m«  ~  5  m*  X  4  w» x«  ~  20 m^n*xfi' 

_3 3  X  20ma:*_  _6()mx*_. 

5na  _  5  n»  X  5  m^ n»  _    25  m^  n^ 
4x^~4x^  X  5  m^n*  ~  20 m^n^icfi* 

Example  2.     Reduce  ^aJ^s^^^is*  a:«!4x-5^  ^'^^  x44^r  +  3 

to  similar  fractions  with  lowest  common  denominator. 

Solution.  The  lowest  common  denominator  is  (x  —  3)  (x  —  5) 
(x  +  1)  (x  +  3),  the  L.  C.  M.  of  the  denominators.  Dividing  the 
L.  C.  M.  by  the  denominator  of  each  fraction,  and  multiplying  both 
terms  of  each  fraction  by  the  quotient  each  by  each,  we  have 

x-1 (a;-l)X(a:+l)(x+3)  (x  +  3)  (x^  -  1) 

x^8x+15~  (x-3)(x-5)  X(x+l)(x+3)  ~  (x  +  l)  (x-5)(x2-9)' 

x+3 (x-f-3)  X  (x-3)(x+3)  (x  +  3)g(x  -  3) 

xa-4x-5~  (x-5)(x+l)X(x-3)(x+3)~(x+l)(x-5)(x2-9)' 

x-5      _  (x-5)X(x-5)(x-3)  (x  -  5)^  (x  -  3) 

x«+4x+3  ~  (x+3)  (x+  1)  X  (x-5)  (x -3)  ~  (x+ 1)  (x-5)  (x^9)* 

Hence,  in  general, 

To  Reduce  Fractions  to  Equivalent  Fractions  having  the 
Lowest  Common  Denominator  (L.  CD.).     Find  the  L.C.M.  of 

the  denominators.  Then  multiply  both  terms  of  each  fraction  by 
the  quotient  of  the  L.  C.  M.  divided  by  the  denominator  of  that 
fraction. 

Kotef :  1.  When  the  denominators  have  no  common  factors,  the  multiplier 
for  both  terms  of  each  fraction  will  be  the  product  of  the  denominators  of  all 
the  other  fractions. 

2.  In  all  operations  with  fractions  it  is  better  to  separate  the  denominators 
into  their  factors  at  once;  and  sometimes  it  is  also  convenient  to  factor  the 
numerators. 

3.  It  will  he  observed  that  the  terms  of  each  fraction  are  multiplied  by  an 
expression  which  is  obtained  by  dividing  the  L.  C.  D.  by  its  own  denominator. 
It  is  not  necessary  to  state  how  the  multiplier  is  obtained  in  every  expression. 


174  ELEMENTS  OF   ALGEBRA. 

Exercise  67. 

Eeduce  to  similar  fractions  with  L.  C.  D. : 

a    m    X    ahe       1       2       5 
b'   n'  y^   mn^    ah'  ac    he' 

m  +  n    m  —  n     n  a    a  —  n      n 

ah  be        a  c  b        in        da 

m  +  2  n    2  m.  —  3  n    5  m  —  n 
Sm     '         6  n       '    10  m  7^  * 

1  x  +  2  x-2 


6. 


x'^-V  x^-V  x-2'  x^-x-2 
m  •—  n    m  +  2n        m^  1 


m  +  n'    m  —  n     m^  —  7i^'    a  +  b'  a  —  h'  a^  +  b'^ 
8m  +  2     2m-l      3m +2 


ft 

m  —  2  '   3  m  —  6 '   5  ??i  —  10 


_  xy  m  —  n 


8. 


Tfix  —  rmy  •\- nx  —  ny'   2  oc^  —  2xy 

m  n  a 

m  +  x'   m^  +  x^'   m^  —  mx  +  a^ 

X  y  m 


x^  —  xy  +  y^'   x^  +  X y  +  y^'   ic*  +  x^y^  +  y^ 
^^  x  —  y  x^  +  ?/2         y        x^  +  y^ 


11. 


x^  +  xy  +  y^'   x^  —  y^'    x  —  y^         5 


12. 

13. 


ALGEBRAIC  FRACTIONS.  175 

b  X 


(a  -f  xf  -  62'   (5  _^  2^)2  _  ^2»   a:«  -  (a  +  6)« 
ate 


(c-a)(6-c)'   (a-c)(c-6)'    {c-a)(c-l) 

a         _  g  X  -1 -g 

Sestioii.    (^_^)(^_c)  -  f(c-a)  X  -l](6-c)  -  (g-c)(6-c)*    ** 

3a         4a  5  am  n  y 

3"=^'   o^'   (a -3)2'   n^'   wrn:'   r=^2 

12  3  4 


14. 
15. 


(2-a;)(3-aj)'  (a:-l)(2-a:)'  (a:-2)(l-^)'  (a:-l)(a>-2) 


1  1 

Suggestion,     (g  -  x)  (3  -  a:)  =  (x  -  2)  (x  -  3)  =  ^*^- 


16. 


17. 

18. 
19. 


(m  —  x)  (a;  —  n)  *    (a:  —  ?w )  (a  —  a;)  '    (a:  —  a)  (n  —  a;) 

1  +  a:  2  -i-  a; 

(l-a;)(2-a:)(a;-5)'    (x - 1)  (2  -  aj)  (3  -  a:)  (5  -  a;) 

a;- 3  a;-  2  g2  +  4  2 

4^=^'   a^+a;-6'   9-6a;4-ar»'   ir2-a:-6' 

^  ar^"*  +  1  A2m_  1 

a:*"-  1'    a^'"  +  42:2.-^_  3'   a^«  +  2  ar»'"  -  3  ' 


79.    Example  1.     Find  the  sum  of  t  ,  -\^  and  -. 

o    a  n 

Solution.      Multiplying  the  terms  of  the  first  fraction  by  dn, 

of  the  second  by  bn,  of  the  third  by  6rf,  and  adding  the  results 

(Arts.  32,  14),  we  have 

a      c      m  _adn      hen      hdm  _adn-\-hcn  +  bdm 
h'^  d'^  ~ii~  h(U'^  bd^'^  bdTk  "^  hdik     ^~  " 


176  ELEMENTS  OF  ALGEBRA. 


m  a 

Example  2.     Subtract  -  from  t 

n  0 


Solution.     Multiplying  the  terms  of  the  first  fraction  by  6,  of  the 
second  by  n,  and  subtracting  (Art.  19),  we  have 

a      m       an      hni      an  —  hm 


h       n       bn       bn  b  n 

2a-3h  ^         Sx-2b 
Example  3.     Subtract  — ^^ irom  — ^ • 

Solution.     Keducing  to  similar  fractions  with  L.  C.  D.,  we  have 

3  a:-  26   2  a  -  3  6  _  6ax  -  4ab      6ax-9bx 
3x  2a   ~~    6ax  6ax 

6  ax  —  4ab  —  (6  a  x  —  9  b  x) 
~"  6ax 

■'J  _      b(4:a-9x) 

~  Hax 

T^.  -11      p    2x  —  my  Zx  —  ny 

Example  4.     Find  the  sum  ot  a  -I '-  and  b 

m  n 

Solution.  Uniting  the  entire  parts,  and  reducing  to  similar 
fractions,  we  have 

/        2x  —  my\       (^      3  a!;  — nwN  ,      (2x—my)n      C3x—ny)m 

[a+ -)+  [h '-)  =  «  +  &+  ^^-^ ^^ 

V  m      /       \  ^      J  '^^  ^^ 

(2x  —  my)n  —  (3x  —  ny)m, 

=  a-i-b-\ '- ■ '- 

mn 

,      (2  n  -  3  m)  a; 

mn 

Note  1.  If  the  sign  of  a  fraction  is  — ,  care  must  be  taken  to  change  the 
sign  of  each  term  in  the  numerator  before  combining  it  with  the  others.  In 
such  case  the  beginner  should  enclose  the  numerator  in  parentheses,  as  shown 
in  the  above  work. 

2^; g  x+2  a;-|-l 

Exampi^e  5.    Simplify  ^2  +  3^  +  2  "  x^--2x-3  "  x^-x-6 ' 


Process. 


ALGEBRAIC  FRACTIONS.  177 

lar-6  x-j-  2  x -\- I 


x2  +  3x+2      x^-2x-3      x^-x-6 

2  (x  -  3)  x  +  2  x+  1 

=  (x  +  l)(x  +  2)  ~  (x  +  1)  (x  -  3)  ~  (x  4-  2)  (x  -  3) 

_  2(x-3)X(a?-3)  (x-t-2)X(x4-2)_  (x4-l)X(a?+l 

"(x+l)(x+2)X(x-3)     (x+l)(x-3)X(x+2)     (x  +  2)(x-3)X(x+l) 

_  2  (x  -  3)^  -  (x  -f  2y  -(x+  1)3  _  13-18X 

(x  +  1)  (x  -h  2)  (x  -  3)  ~  ix+  1)  (x  +  2)  (x  -  3)  * 

Notei :  2.  In  finding  the  value  of  an  expression  like  —  (x  -f  2)*,  the  be- 
ginner should  first  express  the  product  in  a  parentheses  and  then  remove  the 
parentheses  as  above. 

3.  Sometimes  it  is  better  not  to  reduce  all  the  fractions  to  the  L.  C.  D.  at 
once.    Thus, 

14          6  4  1 

Example  6.    -• — 5 -I rTT  + 


x  —  2y      x  —  y      x      x  +  y      x  +  2y 

1  1  4  4  6 

+  iTT-zr-  -  z — -  -  zr-r-7.  +  z 


x  —  2y      x-h2y      x  —  y      x  +  y      x 

x  +  2y  x-2y  4  (x  +  y)         4  (x  -  y)       6 

(x-2y)(x+2y)'^(x-|-2y)(x-2y)     (x-y){x-\-y)     (x+y)(x-y)"^x 

2x  8x  6 


~  x^  -  4  y2      x2  -  y3 

_       2  X  (x^  -  y^)       _      8  X  (x«  -  4  y«)  6 

-  (x«-  4  f)  (x^-y^)  "  (xa  -  y2)  (x3  -  4  y^  +  X 

_        30xy''-6x«  6 

-(x»-4y«)(x«-y2)'^x 

_      (30xt/'-6x»)x  6  (x2  -  4  y2)  (x^  _  yS) 

-  (x2  -  4  y3)  (x2  -y*)x'^  X  (x^  -  4  y^)  (x^  -  y«) 

24  y*  TT  . 

=  x(x«-4y^(x»-t/)-     Hence,  m  general, 

To  Add  or  Subtract  Fractions.  Reduce  to  similar  fractions 
with  L.C.  D.;  add  or  subtract  the  numerators,  and  divide  the  result 
by  their  L.  C.  D. 

12 


178  ELEMENTS  OF  ALGEBRA. 

Exercise  68. 

Simplify : 

2a  —  5       S  a  —  11      b  +  c       a  +  c      a  —  b 
12  a     "^        l8       '    T^  "^  T6  97" ' 

X  2x         ^x^^   xy  xy^  xP"y^ 

^       m       n      fm  +  n     a  +  b\      /m  —  n     a  —  b\ 
'   ab     ac     b  c''    \     n  a    J      \3?i  ^a  )' 

3  +  ^^     4-am       a       /5      4     3\       /I      2      3\ 

4.  + +  ^; +-)+ )• 

n  an         6n     \m     n     xj       \m     n     xj 

5a-b       7a  +  Sb  _  /2a       a  -  b\ 
2b      "^        6^  \b    "^     3  &   y  * 

6.   (^,.  +  ^|)  +  (3m-^)-(4m  +  ^). 

^    a^—bc      ac—b^     ab  —  c^    2  a^—b^     b^—c^     c^—a^ 
i  c  ac  ab    '        a^  b^  (?     ' 

8.   (m  +  n ^  )  —  (  2m  —  371  +  —  V 

\  mxj      \  nxj 

\5  a;       Zy       'omj       \6  x       10  3/      7  m/ 
a  +  &       b  —  c       c  —  a       ab'^  —  b(?  —  c  0? 


11. 


5  ab  c 

1  1        :r4-2      x-2  3 


oj— 5      ^—4'  2;— 2     a;+2'  2m(m— 1)     4m(?w— 2)" 


ALGEBRAIC  FRACTIONS. 

179 

12. 

2a;/i  — 3&71     2a7n  +  'Sbn            1 

1 

'6vin{ia—7i)     3  7/1 7i  (?/i  +  7i)  *  ic^*--4ic  +  4 

"x-'^+x-G* 

13. 

x  +  y                 J^  —  y       '  x—m      X'\-m      x 

14. 

1         m+3            -7           2          3 

...      1      •>  "T  A           2         10> .J.^       " 

2m-3 

1           A  ™2        1  • 

m  n  2  mn  1  (a  +  2xy^ 

m  -i-  71      m  —  71      ir?  —  r?'    a  —  2x      a^  —  S  a^' 

2:  1         1  11 


16. 


xy—  if'     x  —  y     y'    m^—  (n  +  xj^     a^—(m  +  ?i)^ 


x^+'Sx^f+y*     a^xy-\-ii^     2  3?  x  +  y 

x^  —  ^  X  —  y     '  x*—y*     x^+x?y+xif+y^' 

,Q  x  +  4:         ,         x-hS  x+2 


3?+  ox-{-  6      a?  +  6  x+  8      3^+7  x+ 12 

1  mn  m  —  71 

7n  +  71      TTi^  -\-  n^      7n^  —  m  71  -\-  71^  ' 

90  1  1  X  X 

7n  +  X      m  —  X  "  (m  +  xf      (tti  —  x)^' 

21   __i L_  +      ^      4.  _    ^ 


8-8ic      8  +  82:^4  +  40^2^  2 +  2a;* 

22  24a; 3  +  2a:      3  -  2  a; 

9-12a;  +  4ar»      3  -  2  a:  "^  3  +  2a; ' 

00  «  +  &  ?^  +  r-  r  +  ^ 

(6  -  c)  (c  ~  a)  ■*"  (c  -  a)  (a  -  ft)  ^  (a  -  I)  (h  -  c) 


180  ELEMENTS  OF  ALGEBRA. 


24  ^+^y        ,         x-^2y  x^y 


4.{x+y)(:y+2y)      {x+y){x-VZy)     4.{x+2  y)  {x+Z y)' 
^^  he  ac  ,  ah 


(c  —  a)  (a  —  b)  (a  —  b)  (b  —  c)      (b  —  c){c  —  a)' 

26        5(2^-3)  7x         _     12(3a^  +  l) 

•    11(6^-2+ ^-1)      6  2;'^+7ic-3      ll(4:X^  +  Sx+3)' 

x     _     y  x^y+xy^       x^  +  'f  ^  —  y^ 

x^+y^     Q^—i/  x^  —  y^  '  a^—xy-\-y'^'~x^+xy-\-y'^' 

28.  ^"  +  ^*  1-^ 


a;3  +  ^2  _  49  ^  _  49       2:2  _  e  ^  _  7 
29  ^  +  ^' a;  4-  &  ^  +  0^ 


30. 


{a  —  b)  (a  —  c)       (b  —  a)  (b  ~  c) 


Suggestion.     In  finding  the  L.  C.  D.  it  is  better  to  arrange  the 
letters  alphabetically.     Thus, 

&  a  b  a  X  —1 

+  VI — wi — X  =  / — jaT N  +  m — \t:^ — TT7I — \  =  etc. 


{a-b){a-c)  ^  {b-a){b-c)  ~  (a-h){a-c)  ^  [(b-a)  X  -l](b-c) 

x^+2x+4:     oc^—2x-\-4:     x-2a     2{o?-Aax) 3^ 

x-\-2  2—x     '    x-\-a  a^—x^         x—a 

32  1  1  1  .       1        ,        ^ 

(m-2)(x'+2)^(2-m)(^  +  m)'  2a;+l      2aj-l 

4a:              2                  a: -3  a?2 

+  -. T—o\    — ^--o ......+ 


1-4^2'    ^_l_4      :x:2_4^+i6'^^64 
33.   7 iw T  + 


(a  —  &)  (a  —  c)      {b  —  a)ib  —  c)      (c  —  a)  {c  —  h) 


ALGEBRAIC  iRACTIONS.  181 

Q  C 

80.    Example  1.    Find  the  product  of  r  and  ^ . 


a  ,  c 


Solution.     Let  i  =  a:,  and  3  =  y.      Multipljring  both  members 

of  the  first  equation  by  h  and  both  members  of  the  second  by  d  (Art. 
47,  Axiom  3),  we  have  a  =  bxy  and  c  =  d y.  Multiplying  these  two 
equations  together,  we  have  ac  =  bdxy.  Dividing  both  members 
of  this  equation  by  6  d  (Art.  47,  Axiom  4),  gives 

ac  _  a       c 

^  =  xy.     Buta:i/  =  ^X^. 

a      c      ac 

Therefore,  T  ^  ^  =  r-, .    Hence,  in  general,  « 

To  Multiply  a  Fraction  by  a  Fraction.  Multiply  the  numera- 
tors together  for  the  numerator  of  the  product,  and  the  denominators 
together  for  the  denominator  of  the  product. 

Notes :  1.  Similarly,  we  may  demonstrate  the  method  when  more  than  two 
fractions  are  multiplied  together;  also,  for  fractions  whose  terms  are  negative, 
integral,  or  fractional. 

2.  Since  an  entire  or  muted  expression  may  be  expressed  in  fractional  form, 
the  method  above  is  applicable  to  all  casesl    Thus, 

^a     m  ^a     am    a      /     ,n\      a^/>».n\      am  ,  an 

r.  o     Ti'j.u         J     .    -4x3-16x4-15     x2-6x-7 

Example  2.    Fmd  the  product  of  ^   q  ,   o — tt,  tts — r= ^ 

,,  „2      ,  ^  2x2-1- 3x+ 1'  2x2- 17x-f  21' 

and 

4  x*  -  20  X  -f-  25 

Process. 

x2-6x-7  4x«-l 


4x=»-l 

4  x2  -  20  X  -f-  25 

4x«-16x-|-15 

2  x«  -h  3  X  -H  1 

(2x-3)(2x-5) 

"2xa-17x+  21  ^4x2-20x-f25 

(x-7)(x-hl)        (2x+l)(2x-l) 
(2x-|-  l)(x-|-  1)   ^  (2x-3)(x-7)  ^  (2x-5)(2x-5) 

(2x  -  3)  (2x  -  5)  (x  -  7)  (X  -H)  (2x  -f  1)  (2x-l)  _  2x-l 
(2x-|-  l)(x-f  l)(2x-  3)(x-7)(2x-5)(2x-5)~  2x-6' 


182  ELEMENTS  OF  ALGEBRA. 

Explanation.  Factoring  the  iiiuneiators  and  denominators  of  the 
fractions,  multiplying  the  numerators  together  for  the  numerator  of 
the  product,  and  the  denominators  together  for  the  denominator  of 
the  product,  we  have  the  third  expression.  Reducing  the  third  ex- 
pression to  its  lowest  terms,  gives  the  result. 

Notes :  3.  Observe  the  importance  of  factoring  the  terms  of  the  fractions 
first.  Also,  indicate  the  multiplication  of  the  numerators  and  denominators, 
and  divide  both  terms  of  the  fraction  by  their  H.  C.  F.  before  performing  the 
multiplication. 

4.  If  the  factors  are  mixed  expressions,  sometimes  it  is  better  to  change 
them  to  fractional  forms  before  performing  the  multiplication.    Thus, 

/  ah  \  /   _    ah  \  _     a^  iA     _    a^lfi 

V^  a-b)\        a  +  b)~a-b      a  +  b~  a^-b^' 

2  x^  -\-  3  X  4  x^ Qx 

Example  3.    Find  the  product  of  — r—^ —  and  lo^  +  is ' 

Process. 

2a;2  +  3a;       4a:2_6a:_a;(2a;  +  3)       2x(2x  -3) 
4x8        ><  12a;+  18~         4x^        ^    6  (2 a;  +  3) 

_x(2a:  +  3)  X  2  a:  (2  a; -3)  _  2a:-3 

~  4  a;8  X  6  (2  a:  +  3)     ~     12  x    * 

Exercise  69. 

Simplify : 

^     a2       j2        c2      3a3       2h^       7c^     ^  ^ 

^-   Fc^'^c^  aV    4.c^  "^  21  a^""  5  ah'    ^      r' 

Sah^      3«c2      Sad^      Sc-^x^  2()  (^  x 

2. 


4crf        2hd  "^    9hc  '    5a^y-^      9a-^y-^ 

x+1        x  +  2         x-1        Za^-x      10^ 

^'  ^^^=~l^  x^-1^  {x  +  2f'  5         ^2a;2-4ic 


a^  +  3  .r  4-  2      x^^-'Jx-\-\2 ^     m^  -  n^         m^n 
^  _^  9  a;  +  20  ^  :i-2  _|_  5  ^  +  6 '     ^3  _  ^2^       ^3^  ^£ 


X' 


6_ 


ALGEBRAIC  FRACTIONS.  183 


2/6  ^■\-'f  x  +  y 


X 


^-  a^  +  2  3^  ij^  +  1/       s^  -  xy  -{-  1/  ""  a^  -  f 

am        fm      (i\      m^  4-  w*  ^i      f  _!!!: ^    \ 

*   oTw      \a~7wy'     m^  +  n^       \m—n      m  +  nj 

m^-\-mn        nfi  —  n^       a^—(a-^b)x-{-ab     x^—<? 
"'    m^'^z  ^mn{m-\-ny    a^-{a  +  c)x+ ac^  x^-l^' 


m°  — 


m^  +  n^  f..       ^     ^ 

^'  m^  —  m  n  +  n^      m^  +  mn  +  n^       \        m  —  nj 

9.  g  _  ?  +  1  V^2  +  ^  +  1)  •        Suggestion. 

[e-)-i][(s-)-g-(s-)"-0'- 

^^     fx      a      y       h\       (x      a      V      h\ 

10.  ( +  ?--Ix( i  +  -]' 
\a      X      b      yj       \a      X      b      yj 

\bc      ac      ab      a  J       \         a  +  b  +  cj 

a^  •\-  ab  —  ac      (a  +  c)^  —  V^      ab  —  b^  —  be 

a:2--2a;"- 63  ^  a^-  +  3a:"-40  ^  ?M^4^^+3 ' 
r     a^>  y'"*     1>.r  (^'^-?/^"')^  1 


8L    Example  1.    Find  the  quotient  of  -.  divided  by  ^' 

fl  c 

Solution.    Let  x  represent  the  quotient.     Then    t  -^  -5  =  «. 
Since  the  quotient  multiplied  by  the  divisor  gives  the  dividend, 


184  ELEMENTS  OF  ALGEBRA. 


we  have  x  X  -.  =  j^.     Multiplying  both  members  of  the  equation 

d  c       d      a       d  ad 

by  -  ,  we  have  a:X-;X-  =  TX-,  oraj^yX-* 
•^c'  d       c       0        c^  he 

Therefore,  ^^^=:-^X-=^.     Hence,  m  general, 

To  Divide  a  Fraction  by  a  Fraction,     invert  the  divisor,  and 
proceed  as  in  multiplication. 

Notes:  1.  Since  an  entire  or  mixed  expression  may  be  written  in  fractional 
form,  the  above  method  is  api)licable  to  all  cases.    Thus, 

_^a  _  c  _^a  _  c       ^__^c      a         _  a       c  _a      1 a 

^  '    b"!   '   l~l^a~  ~^''     b'^^~l^\~b      c~bc' 

2.  It  is  usually  better  to  change  mixed  expressions  to  fractional  form  before 
performing  the  division.     Thus, 

(_a6\^/  ab  \  _     <fi     ^     b^     _     a'^         a  +  b  __a^ 

"      T+b)   ■   V       aT6/  ~  a  +  6   '   aTl>  ~  ^TTb  ^  ~W~  ~  P  ' 

^.   .,     ar2-14a:-15,      x^-l2x-45 
Examples.     Dmde  ^,_^^^^^    by  ^^--^-_^. 

Process. 

a:g-14ar-15  .  g;2-12a;-45       (x -  15)  (x  +  1)  ,  (x -  15)  (a:  +  3) 
a;2-4a;-45    '    a;2-6a:-27  ~   (x-9)(x  +  5)   ~  (ar-9)(a;  +  3) 

-  (a^-15)(a:+l)         (re -9) 
~   (a:  -  9)  (a:  +  5)    ^  (x  -  15) 
_  (a^-15)Ca;+l)(a:-9)       x+  1 
"  (a:  -  9)  (a:  +  5)  (a;  -  15)  ~  a;  +  5  ' 

^.   .,     a?^      1  ,       a:       1       1 
Example  3.    Divide  ^+ibyp--+-- 

Process. 

\y^      x)  ~  \y^~  y      x)  ~      xy^     '  x  y^ 

_  {x -\- y)  (x^  -  X  y  +  y^)  xy^ 

xy^  x^—  xy  +  y^ 

_x  +  y  __x 


ALGEBRAIC  FRACTIONS.  186 

Exercise  70. 
Divide : 

2  a^  2:1  y  ,        a  x^  y"^  Z  m      ,        2  m 

6  (g  6  -  ^/2)  2  62  2;3_y3         (a;  -  y)8 

•^-     a  (a +6)2    '^^aCa^-feZ)'    a^ -^  f  ^  {x  +  y^' 

x^-f  x-y  ^    a^^xy+ip'  ^  - 1?' 

m8+8^m  +  2'  2?  ^- f        ^  3?-xy-\-y^' 

^    2^:2+13^^15         22:2_^  11^  +  5 

5-   ^^_Q by 


4a:2_9  -^  43^2 


a:2  4-  a:  y  +  ?y2         x  +  y     m 


m 


6.  ;;o ^^-H^  by  — ^^;    -^ ^  by  -  +  -. 

x^  —  xy-)r]rx  —  y      n^       m^         71       m 

yjX-^y      x-y        x+if      x-y        X      y 
'  X  ^  y      X  +  y     ^  X  —  y'     x  -\-  y     ^  y      x' 

x^-^  (a  +  c)x  +  ac         x^  -  o? 

^-  a?»  +  (6  +  ^)  a:  +  6  c     ^  ar»  -  62  • 

a2  4.^>2_^2ft6~c2         «  +  6  +  c 
^-   c2-a2_i,2+2a6  ^  6  +  c-a' 

10.   a;8-^  by  a:--;    a2-62-c2+2&c  by  ^^44^^- 
a:^     -^  a:  -^   a  +  6+c 

11-    ^6  .  ^6    by  -2-7-0;  -e— r  by 


7i6  +  a:«       '    7i2  +  a:2»    fl^6__i      ^    a^^a^^a-l 

^^    x~^  —  x~^.  x~i -{- x~^ 

12.        0^-8       by 


2a:-8  4a^2(x-f-a:-*) 


186  ELEMENTS  OF  ALGEBRA. 


Exercise  71. 

Perform  the  operations  indicated  in  the  following  and  reduce  the 
results  to  their  simplest  forms  : 

7  ic  +  6  .    a:2  +  6a;  \  .  .  a;2  +  lOa;  +  24 

48* 


/x^-7x  +  6^    x'^  +  6x\       x^  +  10a;  + 
^*    \x^-h  3a; -4  ~  x^  -  8 x^J  ^  a;^-  14a;  + 

/x2  -  7  a;  +  6        a;2  +  6a;\        a;'^  +  10 a;  +  24 
Process.     (^^2  +  3  a;  _  4  "^  a;3-8a;2j  >^  a;^- 14a; +  48 

-i  [(^  -  6)  (a;  -  1)  ^  a;(a;  +  6)1        (a;  +  4)  (a;  +  6) 
~  [(a;  +  4)  (a;  -  1)  "^  x'^  {x  -  8)  J  ^  (a;  -  6)  (x  -  8) 


_  fa; -6       a;(x-8)1        (a;  +  4)  (a:  +  6) 
~  [a;  +  4  ^      a;  +  6    J  ^  (a;  -  6)  (x  -  8) 

_  a;  (x  -  6)  (a;  -  8)  (x  +  4)  (a;  -f  6)  _ 
-    (x  +  4)  (X  +  6)  (X  -  6)  (x  -  8)  ~  '^' 


a-l     a  +  1     «2_i 

X : 

a  +  i      a  —  l'a  + 


1'    Va-&~  a  +  &J  *  a2-62- 

\x  +  y       X  —  y       x^  —  y^J        \x  -\-  y       or  —  y^J 

^2  _  ^  _  20        x^-x-2       x^-S6        ^  +  1 
4.    — 5 :t?—  X  -0-7-r. o  X 


x^-2d  x^  +  2x-S^x^-6x  '  x^  '\-  5x 

?/4  a  +  h       x^—3xy  +  2y^  ^  {^  —  vf 

X  7        ; ZTt  X 


'   a^h  +  ah      {x  -\-  y)^  x^  +  y'^  '       ah 

/a-f  Z)       «  —  &\        /«  +  &       a  --  &\ 

max       a^  —  x^       h  c  -}-  h  x       c  —  x    ^   mx 

noy        c^  —  x^       a^  +  a  X       a  —  x       ny 


ALGEBRAIC  FRACTIONS.  187 

8       1      •   PM      I      Mx     ^"^  1. 

'   x-^y       \_2\x  +  y      x  —  y)      x^y  +  xy^J 

x^-\-x-2      aP-^5x+4:  .   fx^+Sx-{-2      x+3\ 

^^'    \6x-62  *  x^^l^J'^ax+a^'  8a^y^^  21b"'^^y'^-^ 
X     6^mH^-a^        (a:  -  2)^     ,    ar^  -  4 
^a'      ,^2-4       ^  8  ?w?i  +  a   *  (ir  +  2)a* 

12.    .,    o   a  X  — =— i-  X  '^ 


14.    (a:*  —  -jjH-fa; J,   by  inspection. 

15-   (p  -  2  +  ^2)  -  (?  -  I)'  ^y  inspection. 

16.     ic^— -g  — sfa; j  \-^ix ),   by  inspection. 

6flg^>g   ,    r3a(m-7i)   .  5  4(c-a;)   ^      c^  -  a^    ^ 
m+n   *    I7{c  +  x)    '  I  21afe2    *  ^(m^-n^ij 


188  ELEMENTS  OF  ALGEBRA. 

82.   A  Complex  Fraction  is  one  having  a  fraction  in  its 
numerator  or  denominator,  or  both  ;  as, 

n'        ,  m  * 
n  +  — 
n 


Example  1.    Eeduce  -  to  its  simplest  form. 

c 

d 

Solution.     A  complex  fraction  may  be  regarded  as  representing 

the  quotient  of  the  numerator  divided  by  the  denominator.     Hence, 

a 

h       a  ^  c       a       d  _  a  d 

cj^l^d'^'b^'i^'bc' 

d 

h 
a  —  - 

Example  2.    Reduce  f  to  its  simplest  form. 

m 

Solution.     Since  the  divisor  is  m,  we  have 
6 

\c  —  b      m      ac  —  h       1       ac — h 


=  [a  —  ~\  -^  m  =  - 


x-  = 


I  c  m         cm 

I 
I      m  ,  m 


Example  3.    Reduce  ~  »  T  »   and  —  to  their  simplest  forms. 


^  1  m  n       n 

Process.     — =zi-f-  =  i  x  —  =  — 

m  n  mm 


m               1  n 

Y  =  wi-r-  =  m  X  ^  =  mn. 

n 

1 

m       1       1       1  n       n 

-r-=~"^~  =  ~  X7  =  -.     Hence,  m  general. 

I       m       n      m  I       m                  '       & 


ALGEBRAIC  FRACTIONS.  189 

To  Simplify  a  Complex  Fraction.     Divide  the  numerator  by 

the  denominator. 

Example  4.    SimpUfy  "*' "  ^^' ~ '"' + '^^ 


m  +  n      m  —  n 


m  —  n      m  +  n 
Process. 


m  -j-  n      m  —  n    ~     (m  -h  ?i)2  —  (771  —  n)^    ~  4  wm 


m  —  n      m  4-  n  (m  —  n)  (m  +  n)  (m  —  n)  (m  +  n) 

4  m^n*  (m  —  n)  (m  +  n)  in  n 


X 


(m«  -  n2)  (m^  +  n^)  '^  4  mn  m*  +  n^ 


Example  5.    Simplify  ^ ^ iiJ2 


x-y      x+y 

Solution.     Multiplying  both  terms  by  (x  — y)  (x  +  y),  the  L.C.D. 
of  their  denominators,  we  have 


2x1/ 


Notes :  1 .  In  many  examples  it  is  advisable  to  multiply  both  terms  of  the 
fraction  by  the  L.  C.  D.  of  its  denominators  at  once. 

2.  If  the  terms  of  the  complex  fraction  are  complicated,  the  beginner  is 
advised  to  simplify  each  separately. 


mp 


Example  6.    Simplify  ^'+(^+^03:+mn      x^  +  {m+p)x  +  mp 

X*  +  (n  +  /?)  X  +  n  p 


190  ELEMENTS   OF  ALGEBRA. 


Process. 


mp  mn  mp 


v'^+(m-^n)x  +  m7i       x^-h{m+p)x  +  mp  _  {x-\rm){x  +  n)       (x+m)(x+p) 
n  —  p  ~~  n  ~-  p 

x'^+ {n+p)x  + np  {x  +  n){x+p) 

m  n  {x-\-p)  —  mp  (x+n) 
_    {x-\-7ri)  {x+n)  {x+p)    _  inx  (n  ~- p)  {x  +  n)  (x+p) 

n  —  p  ~  {x  +  m)  {x+n)  {x+p)  n—p 

(x  +  n)  {x+p) 
rax 


X  +  m 


Example  7.     Simplify  T+~x' 

1 '- 


l-x  +  x'^  + 


a:2_  1 

1  +  X- 


X 


Solution.    Begin  with  the  complex  fraction x^—\ '   '^^"®» 

a:2  _  1       x+l  ,  X  x'^ 

i  +  x-——  =  -^,  and ^5^n[  =  rri-   ^'"^"^^^^y 

i  +  ^-^r- 

l+x^  (l  +  x^){l  +  x)         ,^  l+x^ 
• ^=      ,:8  +  ^2+i      ,and  1--  x^ 

l-X  +  X^+  ——r  l-X  +  X^  + 


x^  +  x^+l 
Therefore 


'  1  +x^ X 

1 X  x^  +  x^+l 

1    -  X  +  X^  +   -o 7 


1  +x-  ——- 

X 
=   -  (X8  +  X2  +  1). 

Notes :  3.  A  fraction  of  the  form  in  Example  7  is  called  a  Continued 
Fraction. 

4.  To  simplify  a  continued  fraction,  the  student  should  always  begin  with 
the  last  complex  fraction  in  the  denominator. 


ALGEBRAIC   FRACTIONS.  191 

Exercise  72. 
Keduce  to  their  simplest  forms : 

a;  +  6  +  ^      1  +  -      a:  +  -      xy 

^  j:  —  6  wt  c  m  n 


1      '     6        ^  '  mm  n       ^ 

X-6  + 1  a;+-     +  271 

X  +  o     m  n        X 

,    .    6^     1   .    1      ni  b  m  +  n 

a-\-b  +  ^     -  +  —     . 

^    a      n m      n  m  4:mn 

0      n      mm  n  S  m^n 


a;+la;-l      ar8-17a;+72 


2  x-1  ./J  +  1 ,  2:2  ^  22  a;  4-  120 
^^  +  1  x-V  x"  -  21  a:  +708 
x^^      iiTTl      a?»  +  18  a;  +  80 


1  +  « 0  4a  6         \a^;       aj\x       a) 

l  +  a6  2a6  a;  +  a 


J,  +  J_.JL 


-    mn      mp      np      f   x      ,  1— A_/    x        \—x\ 

'    m^-(/i  +  pF^  Vh^   ~^y   \i+^"""^/ 


6. 


m  71 
1  1  1 


^      1     '  1     '  ^-1     • 

«.'  + J  1 X — 

x  +  -  1  +  i  ^  +  — ^- 

a;  a;  a;  —  1 


192 

7. 


ELEMENTS  OF  ALGEBRA. 
1  X-2 


1  + 


l  +  a;  + 


'1^ 
1-x 


x-2 


x-1 


x  + 


y 


X 


X        1    ,    1'        _L 
y^       y       X 


x  —  2 

2  xy 

~  (^-  +  yf 


1+ 


x+  1 


d  + 


m 


1  + 


^,y 


(^'  -  yf  X 


ax 
a?      «2 


X' 


X 


X 


X  x^ 


ir 


+  -0        -0+-+1 


a  a? 


2/^       ^ 


X  —  y      x"     y^ 
y     ^ 


10. 


11. 


w?  +  n^         2  m 
m^  —  ii^  '   7?<  +  n 

~m  n  —  m^  ^   m  -{•  n 
(m  —  iif'    '   'IV  —  n_ 

m  —  n 


71  + 


m  —  n 
1  +  m  n 


{in  —  n)  n 


1  -\-  mn 


m 

m 

—  n 

1  - 

•  m  n 

1  - 

(m  — 

n)m 

1  —  mn  ^ 


fm       n\ 
\n       m) 


83.    Example.     Find  the  third  power  of  t; 


Solution.     Since  an  exponent  shows  how  many  times  an  expres- 
sion is  taken  as  a  factor,  we  have 


'aJ^\ 


{aJ^y 


J^nj    -  6"    >^    &«    ><    ftn   -    (hny   ~   h^n  ' 


Hence, 


To  Find  any  Power  of  a  Fraction.     Raise  both  terms  of  the 
fraction  to  the  recjuirecl  power. 


ALGEBRAIC  FRACTIONS. 


193 


Exercise  73. 
Expand,  by  inspection,  the  following: 

^(-•-^)'  m'  [-G)'^a)T 

/  2aHixiy'      fix  +  ;/fy      rm  (x  -  y)!^ 

r(r+,,)(x-,,)y    r(«-i)-»(«-5)n«    /a^a-5)i\w 
*•    L     "«  +  »     J'    L   (2«+3t)i  J'    ^      ^      J 


«-C4?)U©-©T= 


+  1 


arff  X 


wi 


ai 


Jx 


7«^ 


84.    Example  1.    Find  the  rth  root  of 


6" 


Solution.  Since  the  rth  power  is  found  by  takinp^  the  numerator 
and  denominator  r  times  as  a  factor,  the  rth  root  is  found  by  taking 
the  rth  root  of  each  of  its  terms.  The  operation  is  indicated  by 
dividing  the  exponent  of  each  term  by  r.     Thus, 

13 


194 


ELEMENTS  OF  ALGEBRA. 


5«4- 


Illustration,     y  243^  =^  ssTs^^isTs  -  3-p  •     ^®^c®» 


To  Find  any  Root  of  a  Fraction.     Take  the  required  root  of 
each  of  its  terms. 

Example  2.     Find  the  square  root  of^2-"2  —  aa;+j  +  —  +  ^- 


Arranging  according  to  powers  of  a,  we  have 


X      a 


Process. 

First  term  of  the  root  squared, 

First  remainder, 

First  trial  divisor,  a^ 

a 
First  complete  divisor,    «  +  - 


-  times  first  complete  divisor, 


Second  remainder, 
Second  trial  divisor, 


4  +  ^  +  ^^-"^-^  +  ^^ 


x^  a^ 


a2  + 


2a 


X 

2  a      X 
Second  complete  divisor,  a'^  +  -— —  - 


—  -  times  second  complete  divisor, 


a8 

+ 

a2 

X^ 

X 

rc2 

-ax- 

-2  + 

a2 

-aa;-2  + 


a2 


Note.     If  we  take  —  |-  for  the  square  root  of  ^ ,  we  shall  arrive  at  the 


result  —  -p: f-  -- . 

2       a;       a 


ALGEBRAIC  FRACTIONS.  195 

Exercise  74. 

Find  the  values  of  the  following  expressions : 

1-   Vy^286'  V        a^       ^  V        343  ar^*   ^     ^243  ^.25"  j  • 

fni^n^\^      (      Z2a^\\      (       (SA.m^n^x^\\ 
\  a^^    )     '    \        b'^    )   '    \     12oaH^7/y  ' 

Find  the  square  roots  of : 


^,  ..       ^^    .    4    .    ^^^  .   ^ 


Miscellaneous  Exercise  75. 

Reduce  to  lowest  terms : 

h(b-ax)  +  a(a  +  bx)     2^-9  a^  +  7ix^  +  9 x-S 
(b-axf-h{a  +  bxf'    a^  +  Ta:^- 9  ^  -  7a;+ 8  * 

21  x^  y^  -  S5  i/^z- 12  x^z  +  20  xyz^ 
l8x^z^-21x^y-S0y2^+S5xfz' 

40r^y*-?>2  2^yz^-5i/^z^  +  4xs^ 
40:^28- 36a:*22'-5/2  +  4oa^2/8* 


196  ELEMENTS   OF  ALGEBRA. 

^3~6a;2-37;r  +  210^    a;4»+10 ^""^ 35 rg2"+  50 a;^+  24 
■  0^+4x^-^7  X-2W        ic3»+9^2n_|.26a:"  +  24 

Find  the  values  of : 

-_^2a;2/^2;^,                .            -           ^a  —  ic 
0.   o  x^  -] o  when  x  =  4,  v  =  -k,  z  =  1:   r: 

when  2;  = 


a  +  b 


„    x^  +  y''^  —  z^  +  2xy     .  ^  . 

^    2;  —  a       ,x'  —  6     ,                   ft2         X  X 

7.  — ^^ when  x  = 7;    -  + 


b  a  a—b     a       b—a       a+b 

a^  (b  —  a) 
when   X  =  ,  '    , — ^  - 
b  (b  +  a) 

,  ^        ...  ^  —   2  a  +  b  a  +  b 

8-    < ^    -       +    a-2b      ^^^'^     ^  ^   ~2~  • 


(X  —  «V       x 
X  —  b  )         X 


9.   i7^[_r  +  V ^ 2  J    when   x=^,  ?/ =  1. 


2ab-\-2bc^-2ccl-^2ad  '  y(a_c)(J+c)+6(-c-&)(a-2&) 
when  «  =  3,  5  =  1,  c  =  -  2,  t?  =  6. 

g^  +  g c  4-  &2  ^  {4^ai^yij^  ^^\ c 

^2  —  «  c  +  52        V'4a&  — 52-2^  ~  a  +  6  +  cH-fZ 
when  a  =  4,  &  =  3,  c  =  1,  c?  =  7. 

Divide : 

12.  !»J-^by™-x;^+gby?+2^. 


ALGEBRAIC  FRACTIONS.  197 

1   ,               \     t:^      m^  .      X      m 
13.   m* i  by  i?i ;    -4 r  by  -  H 

14   a^  +  ^byx  +  -;    a;3  ^  ^  +  ^  _  _  by  ^  -  -  . 
15.  a2_j2_,2_26cby  ^44^'- 


16 


•  ^+i-K^^-^0-''(^'^-^)'^^-'^ 


Factor : 

Simplify : 

20.   3x-{y  +  [2..-(y-.)]}  +  i  +  |^^. 

a;— 1      x—\  a;  +  3      a:  +  3 

"3"*^^^  .  7     ~^T4 

^■^-    a;  +  2      a; 4- 2   *  a;— 2      x—2' 

'~T'^x^  3     '^0^1 

a-\       l-\       c-\ 

3a5c  a h c 

'   be  -h  ac  —  ab  la.1       1 

a       6       c 


198  ELEMENTS  OF  ALGEBRA. 

23  ^  ,  ^  I  ^^  +  ^        , 


9^>2-(4c-2a)2     Uc^-{2a-uy^     4.a^-{U-4.cf 
'    (2a+3&)2-16c2  +  (36  +  4c)2-4ti2  +  (2a+4c)'^-9&2 

\in/'        j\m—n      J     \n^       J  \m^-\-m7i+n^       J 

h 

a  + 


1        X  -{-  a        1        X  —  a              i_L^ 
27.  ^IZ+f!  +  5^Z±Z;   ^  X  («^-5«). 

1        a  +  a;        1        a  —  x'  h  ^  ' 

a         0^  +  Q(^        ^         <x2  _}_  ^2  ^ 

h 

a  —  h  —  c  h  —  c  —  a  c  —  a  —  h 

a^—ac  —  ab  +  bc     b'^—ab  —  cb+ac     c^—bc—ac+ab 


X  X 

11  1 


30. 


2/  +  r 


ALGEBRAIC  FRACTIONS.  199 


31. 


/3  2: +  3:3X2 
yi-f  32:V 


9^  _  33  -  ar^  3 


^       32:2+1  ahc 


a:3_3a:'^  2:^        (a^-a:)^       hc^  ac      ab 


S-a-b  +  c      62     a-16-2      a^b-^       (a-^b-^y 


a  ^b  —  c     *    a^       ab~^ 


aH~^       /a-2  6-^Y 


'^2:'"+?/"        '^2^^'"+?/2'*      a-U^       &Ki        ai(^ 
2:-+y"  x^+y^     jic-i 

10  2; 7/ -3^2+  10a:-3y  _^    lQ2:-3y        _3_ 
15i/^+102;//2  +  :30y+202:y  '   452/  +  302:y  ■*"  y+2  ' 

-  (f-)(i^r')H"-')(Sa?-) 

35   ^  +  ^''  ^  +  ?^^\.f  +  y')  ■  \ff     x) 

(x  +  y)^  —  xy 
(x  —  yf-\-xy 

r    g<-?/       ^  rt2+a?/1    To^-oV     g^-2qg.v+fl27/n 


37.       *  1 


Sax  —  5b  y      ax  '^by'^ 

1        ^""3a2;-2fey 

38.  -i-^+-l-+         1 


1  +  i—        1  +-T—        1  + 


-t  z  X  -\-  z  X  -{■  y 


200  ELEMENTS  OF  ALGEBRA. 

39.    (a  +  2>  +  .)^-  +  ^  +  -j- ^^^ 

(c  -  a)  {a-h)'^  {a-  h)  {h  -  c)'^  {h  -  c)  (c  -  a)' 

fa^—ij^      lOx^—lSxy—Sy^  2a:^  +  xy^+xy+y^\ 

\x^—jf       l<)oiP'—Zxy—y^  2x^  +  xy—%f'      ) 
xy  —  y'^— 2x+2y 
~              2a;-?/ 

(a  +  hf  -  6-3       {h  +  cf  -  gg       (ci  +  c)3  -  63 


44. 
45. 


(a  +  i)  —  c  h  -\-  c  —  a  a  ■\-  c  —  h 

11  1 


a(a  — &)(a  — <?)       b(b  —  a)(b  —  c)       abc 

2a  +  n  a  +  b  + 71  m  +  n—a 


am+ab—bm—o?     ab+bm—am—b^     m^—bm—am+ah 


46.     -7 rc^ X  +   TT, TT^ -X  + 


a(a—b)(a  —  c)       b(b  —  a){b  —  c)       c{c—a)(c—b) 

Queries.  Why  does  changing  the  sign  of  one  factor  of  either  term 
of  a  fraction  change  the  sign  of  that  term  ?  Will  it  change  the  sign 
of  the  fraction  ?  Why  1  When  the  denominators  have  no  common 
factors  why  multiply  both  terms  by  the  product  of  the  denominators 
of  all  the  other  fractions  1  Why  does  the  process  of  reducing  to 
forms  having  a  common  denominator  not  change  the  value  of  a  frac- 
tion ?  How  prove  the  methods  for  addition,  subtraction,  multiplica- 
tion, and  division  of  fractions  1 


FRACTIONAL  EQUATIONS.  201 


CHAPTER   XV. 
FRACTIONAL    EQUATIONS. 

85.    Example  1.    Solve  -j^  -  147^315  =  "gT 30~ 

J_ 
■*"105* 

Solatioii.     Multiplying  each   member  by  210  (the  L.  C.  M.  of 
15,  21,  30,  and  105),  transposing  and  uniting  like  terms,  we  have 

— j =  5  +  30  X.  Multiplying  each  member  of  this  equa- 
tion by  X—  1,  transposing  and  uniting  like  terms,  we  have  25a;  =  100. 
.-.  x  =  4. 

Proof.     Substituting  4  for  x  in  the  given  equation,  we  have 

6-5X4      7-2X4"  _  1+3X4  _  10  X  4  -  11         1 
15  "14(4-1)  ~         21  30  "^105* 

or,  —  ^^  =  —  ^yV'  which  is  an  identity.     Hence, 

To  Clear  an  Equation  of  Fractions.     Multiply  each  member 
by  the  L.  C.  M.  of  the  denominators. 

o      c,  1      2x+U       2fx-l       x-l 

Example  2.     Solve  =— ^  -  ^ =  -^  • 

5  50  z  —  10        2^ 

Proceas.     Multiply  by  5,  2  x  +  1^  -  loT^  =  2  x  -  1. 

Transpose  and  unite,  -  r^ — — ^  =  —  2^. 

Clear  of  fractions,  -  (2^  a:  -  1)  =  -  25  x  +  5. 

Transpose  and  unite,  22.6  x  =  4.    .*.  x  =  ^W* 


202  ELEMENTS  OF  ALGEBRA. 

Note.  In  solving  a  fractional  equation,  where  some  of  the  denominators 
are  simple  and  some  are  compound  expressions,  it  is  better  to  multiply  each 
member  of  the  equation  by  an  expression  which  will  remove  the  simple  denom- 
inators tirst,  then  transpose  (if  necessary)  and  unite  like  terms.  Similarly 
remove  the  compound  denominators  of  the  resulting  equation. 


Exercise  76. 

Solve  the  following  equations  : 

■    X  ^  Vlx~  24'        2  '6x    ~       ^      ~    'Ix    ' 

6a;+13_  32:4-5       2x     2^-5        x-?>    _4a:-3        ^ 
15         5x-25  ~  5  '        5     "^  2i^=^  ~    10         i^- 

9  2^+5      8^-7_36^+15      lOJ 
14      ^6^+2"       56       "^l4" 

4a:+3      73;-29  _  Rrr+19      3,x+2_2a:-l  ^  a; 
9      "^5:^-12"       18  .  '         6  3a:-7       2* 

^    \^x-Tl      ^         I+I62;       ,.        101-642; 
^-  ^9^^6^+2^  +  -^4-^^^^-— 24 


6. 


18  2^+10      72  a;  +  30  _  20.5       16  2;  -  14 
42  168       ~   42         18aj+  6 


1         2     _2;+2     4(2;4-3)_82;+37      72;-29 
^-   2"^'2:+2~    22;   '  9        ""       18  52;- 12 

^    2  2:  +  8r}  _  13  2;  -  2    ,   ^  _  7^  _  3^+16  ^ 


9  17  2^-32  '   3~  12  36 

g+  1       22:-4      22;-  1      x-2     x-4: 
T5~      72;-16~       5      '       .05        -0625 


FRACTIONAL  EQUATIONS.  203 

86.  Frequently  it  is  better  to  unite  some  of  the  terms  before 
clearing  the  equation  of  fractions.     Thus, 

X 

^^  ~  3      16x4-  4.2  23 

Example  1.    Solve    -^^y  +     g^.^^     =  ^  +  ^Tl * 

X 

^^~3        23         16X+4.2      ^ 
Process.    Transpose,  — ^  -  ^q-[  +  -3^+2"  =  ^- 

,,  .     ,,  ^~3      I6X+4.2      ^ 

Unite  like  terms,  jTTj"  H — g     ,  «    =  5. 

Free  from  fractions,  4+V-x-a:^+16x2+20.2x+4.2  =  15a:^25a:+10. 

1.6x 
Transpose  and  unite,  — g—  =  1.8. 

.-.  a;  =  3f. 

Example  2.    Solve  -^—^  -  jqjg  "  ^a^Ti  =  0- 

Process.     Multiply  by  x^  -  4,  (x  +  2)  -  (x  -  2)  -  (x  +  1)  =  0. 
Simplify,  -x+3  =  0.     .-.  x  =  3. 

Notes :  1.  If  a  fraction  is  preceded  by  the  —  sign,  in  clearing  the  eqiiation  of 
fractions,  care  must  be  taken  to  change  the  sign  of  each  term  of  the  numerator. 
In  such  case  it  is  convenient  to  enclose  the  numerator  in  parentheses  before 
clearing  the  equation  of  fractions. 

2.  The  student  should  be  careful  to  observe  that  he  can  make  but  two 
classes  of  changes  upon  an  equation  without  destroying  the  equality  : 

I.  Such  as  do  not  affect  the  value  of  the  members. 

II.  Such  as  affect  both  members  equally. 

Thus,  in  the  above  process,  the  first  operation  affects  both  members  equally; 
and  the  second,  that  of  uniting  like  terms,  does  not  affect  the  value  of  the 
members. 

4  2  5  24 

Example  3.    Solve  ^-j^  -  ^^^^  =  ^-^  -  ^-^ . 

Solution.    Transposing,  ^  -  g^  =  ^  -  2F+2 ' 


204  ELEMENTS  OF  ALGEBRA. 

Simplifying  each  member  separately,  we  have 
3        _        11  1 


2  (x  +  3)       2  (a:  +  1)  '  "^    a;  +  3  -  2  (a:  +  1) 
Clearing  of  fractions,  we  have  2  (a;  +  1)  =  x  +  3.     r.  x=l, 

-c^  A      oi       ^  —  4      x  —  5      x  —  7      x  —  8 

Example  4.     Solve  r 

x-5      x-Q      x-S      x-9 

Solution.     Reduce  the  fractions  to  mixed  expressions, 

1  1  1  1  ^    ,      .  , 

or  ^-3-^  -  ^^^^  =  ^-jg  -  ^^^  •  Reducmg  the  terms  in  each 
member  separately  to  common  denominators  and  adding,  we  get 

-  (a:-5)(a;-6)  ="  ~  (a:  -  8)  (a;  -  9) '  ^^^^""^  *^^«  ^^"^^^^^  «f 
fractions,  we  have  —{x  —  8)  (^  —  9)  =  —  (x  —  6)  (x  —  6).  Simplify- 
ing, transposing,  and  uniting  like  terms,  —  6  a:  =  —  42.     .'.  x  =  7. 

(2  a;  +  3)  a;      J^ 
2a;+  1      "^  3a; 

dx  A-  3^  X 

Process.    Reduce     ^^  ,   i      to  a  mixed  expression, 

Transpose  and  unite,  -  ^j:^  ==  -  3^  ' 

Clear  of  fractions,  — 3a;  =  —  2a:- 

Therefore,  a;  =.-  1. 

-n.  r.  ,       5a;-64      2a:-ll      4a;-55 

Example  6.    Solve 


a:- 13  x-Q  x-14:       x-1 

Process.     Reduce  the  fractions  to  mixed  expressions, 


FRACTIONAL  EQUATIONS.  205 


Simplify  each  member  separately, 

7 


(x  -  13)  (x  -  6)  ~  (2:  -  14)  (x  -  7) 
Divide  by  7  and  clear  of  fractions, 

x2  -  21  a;  +  98  =  a?2  -  19x  +  78. 
Therefore,  x  =  10. 

Exercise  77, 
Solve  the  following  equations  : 

12  1  29       x  +  4         x+6 


a;    '    12a;       2.4*    32;-8       3a:-7 

3a:+l        a;-2      6a:+l        2  a;  -  4       2a;-l 


3(a:-2)       a;-l'        15  7a:-16  5 

x-\-25  ^2x  +  75  5  4       _       3 

a;-5  ""  2a:-15'    1  -  5  a:  "*"  2  a;  -  1  ""  3a:  -  1 


6a;+8       2a;+38_         x^^x+1       a^+x-{-l  _  ^ 
2a:+l        a;+12    ~     *       a;-l      "^      a:  +  1     "  ^^' 

^7_2^--15  1  J 2 1__ 

a:+7      2x'-6"*"2a;+14"~    '1-a:      1+a:      l-x^~^' 

3  30  3.5 


4  -  2  X-       8  (1  -  a;)       2  -  ./;       2  -  2  a; 

6^-7i  l  +  16a:         ,        121-80. 

^'  13-12a:+'^^+        24       ^  ^^^  3 

a;— 1      a;  —  5      a;  —  4a;--2 
aj  —  2      a;  —  6       a;— 5       a*  —  3 

5a;-8       6a;-r44       10a;-8a;~8 
'     a;-2  ar  — 7  a;  —  1     ""  a;  —  6  ' 


206  ELEMENTS  OF  ALGEBRA. 


x-l       x+1  _2(x^  +  4x-tl) 
^^ •  x-2'^  x  +  2~         {x+2f 

, ,     30  +  6  rr       60  +  8  a^       ^  .  48 

11.    ; —  H ; — rt —  =  14  + 


X  +   I  X  +   O  X  +   1 

.6a:+.044  .5^--.178_  .3a^-l  _  .5 +  1.2 a; 

■^^'  A  .6         ^-^^^   .5x-A~   2x^1 

2x-Z  Ax -.6       1-lAx  _  .7{x-l) 


13. 


.3ic-.4       .06;:c-.07'      x  +  .2  .1  -  .b  x 


87.   A  Literal  Equation  is  one  in  which  some  known 
number  is  represented  by  a  letter;  as. 


X  X 

Example  1.    Solve  — f- 


m      n  —  m      m  -jr  n 

Process.     Clear  of  fractions,  x {n"^ —m^)+x (m^ +mn)  =  m^{n- m) 

Unite  like  terms,  {n^  +mn)x  =  m^{n  —  m). 

m^(n—m) 
Divide  hy  n(n  +  m),  x  =  ^^^^^_^^^y 

Example  2.    Solve  (x-m)  (x-n)  —  (x-n)  {x-a}  =  2(x-m)  (m-a). 

Process.     Simplify,  transpose,  and  unite, 

Sax  —  3mx=:  —  2m^+  2am  ~  mn  -\-  an. 

Factor,  3  (a --m)x  =  (a  —  ?«)  (2m  +  n). 

2m  +  n 
Divide  by  3  (a  —  m),  x  = ^ —  • 

a2-3&x         ,„      ,  6  6a:-5rt2 

Examples.      Solve    ax ab^  =  ox-\ -^ 

a  2a 

bx  +  4a 

4 

Process.     Clear  of  fractions,  simplify,  transpose,  and  unite, 

4a^x-3abx  =  4a^b^-  10  a^. 
Factor,  a  (4 a  -  3b)  x  =  2  a^  (2b^  -  5). 

Divide  by  a  (4  a  -  3  6),         •  x=     ^\_.^f^     ' 


FRACTIONAL  EQUATIONS.  207 

_,      ax  ~b      bx  —  a  a  —  b 

Example  4.    Solve ,  .  —  l^  ,  ,.  =  /„  ^  ,  h\  /k ^ ^  „\  ' 

ax  -{■  0      ox  +  a      {ax  +  o)  {ox  +  a) 

Solution.     Reducing  the  terms  of  the  first  member  to  mixed  ex- 

/  2b   \       f  2a   \  a-b 

pressions,  we  have  [l  -  -^-^ j  -  [l-  ^^^j  =  (^^^^^^^^^^^  • 

Uniting    like    terms    and    reducing  the    fractions    to    a    common 

denominator,    adding    and    factoring    their-  numerators,    we    have 

2(a-{-b)  {a-b)x  a-b  ^,,      .         ^  ,       . 

7 .   iv  /I. — ; — 7  =   7 ,   .V  .. — ; — ; .      Clearing  of  fractions, 

{ax  +  b)  {bx  +  a)         {ax  +  b)  {bx  +  a)  "  * 

2{a  +  b)  {a  —  b)  X  =  a  —  b.     Therefore,  x  —  ^  .        .  v  • 

Notes:  1.   Example  4  may  be  solved  by  clearing  tlie  equation  of  fractions. 
The  solution  is  presented  as  an  expeditious  method. 

2.  If  the  student  cannot  readily  discover  a  special  artifice,  be  should  clear 
the  equation  of  fractions  at  once. 

3.  Known  terms  are  called  absolute  terms.    Thus,  in  the  equation  mx^ 
•\-  nz  -\-  a  =  0,  a  is  called  the  absolute  term. 

a  -i-  b         a  b 

Example  5.    Solve ;  =  0. 

x  —  c      X  —  a      X  —  b 

Process.    Clear  of  fractions, 

{a-\-b){x-a){x-b)-a{x-b){x-c)-b{x-a)(x-c)  =  0. 

Simplify,  transpose,  and  factor, 

x{ac  +  bc  -  a^-b^)  =  ab{2c  -a-b). 

Tx.  .,    ,  ,  «      .„  ab{2c-a-b) 

Divide  by  a  c  +  6  c  -  a*  -  62,  x  = 7^; 5 — A* 

^  '  ac  +  bc  —  a^—b^ 

_fl&(a  +  6-2c) 

°'  ^~a2  +  63-c(a  +  6)' 

Exercise  78. 

Solve  the  following  equations : 

10.  e  a      6        1 

X      a       ^x 

2.   \0hmx  —  ^an  =  2am  —  hhnx\ = r* 

a       X      X      0 


208  ELEMENTS  OF  ALGEBRA. 

^    7n?      n      4:71^      m      a  h  „      ,« 

X        A         X  4:      ox       ax 

4.-|.(.-«)-(^-±^J=^(.-|). 

5.  ^^  -  ^^  +2  =  0;   (x-a)(x-h)  =  {x'-a-hf. 

^    a{b^x  +  a^)  ax^     2fx      A       Zfx        \ 

6.  -^— V ~  aca;  +    ,-;    --  +  1)=--  —  1). 

hx  b        S\a        )      4\a        / 

3      ah  —  x^     4:X  —  a  c     x^  —  a      a  —  x     2  x     a 


^                 ^   ^                  ^       ^^                  w^.^                         l^                 ^                 ^ 

'   c          hx               ex      '        hx             h 

h 

^    X  —  m       0?  —  mx  —  V?      ^             n? 

0                          L                                          —  1      • 

VI               mx  —  n^                  mx  —  n^ 

Miscellaneous  Exercise  79. 

Solve  the  equations  : 

X      07+1       ^  —  2x'^ac      hc_  , 

9  ^  ~1>  1  -  9aj'    h~x~'^~^  '^    ' 

ax+h  Sh      ^a^x^  +  h^ 

ax  —  h       a  X  +  h~  a^x^  —  b^' 

X  X  ■\-  \  _x  —  ^       X  —  ^ 

'  x  —  2       X  —  1       X  —  ^      X  —  1 ' 

2(2a;+3)  6  bx+\ 


4  1 


63-9^        1-x      2% -Ax 


FRACTIONAL  EQUATIONS.  209 

5  1         I         1 1 0. 

a  (6  —  a;)       b{c  —  x)      a(c  ^  x) 

6.   (2a:--^)rar+^^  =4a:^^-a:Vj(a--4a;)(2a  +  3a:). 

17      __  .    _   105  +10a;  _  .^ 
^*  a;  H-  3        ^  ~       3  a:  +  9 

8.  (.^3)^_^^^)  =  7.-(3.-?i^)). 

a?— a     a  +  a;       2aa;_  1     ^    1    __  a  —  h 

'   a—b     a -{•  b     a^—l^~     *   x  —  a     a;— 6      x^—ab 

10.    3— + — j —  =  2  a;;   -=c(a  — ^))  +  -. 

a;  —  la;  41  x        ^  ^      x 

_     a:+  2   ,   a:-  7       a:  +  3       x  -  ^ 

11. h  ■=  —   — — z-  =  7  • 

a;  X  —  o      X  +  1       X  —  4: 

135  a;  -  .225       .36       .09  x  -  .18 


12.   .15  a; + 
13. 


.6  ~   .2  .9 

x—a  x—a—1  x^b  x—b—1 


a;-a  — 1       a?  — a  — 2       x—b  —  l       x—b  —  2 


^  .     SO  a  —  bx       9  n  —  ax      6  m  —  nx 
14.   = 5 ^ =  0. 

,^    4m(a2-5./2)      ^  5  m  (J^  -  2  a;) 

8a:  4 


X  —  np      X  —  mp      X  —  mn 

16.  p = p. 

^  m  n  V 

14 


210  ELEMENTS  OF  ALGEBRA. 


3  &  (a;  —  a)       a;  —  &2  ^      &  (4  a  +  c  a?) 
5  a  15  6    ~  6a 


mx  —  n       mx  -\-  n 

c^  —  Sdx(P+2cx  X X 

'   c^+odx      d^—2cx~~*         m       ~~        n 

n  ',n 


m       x       7n(x—m)      x(x  +  m)         mx 
x       m      x{x  +  m)       m{x—m)       m^—x^ 


Queries.  Upon  what  principle  is  an  equation  cleared  of  fractions  ? 
How  is  it  done '?  Why  change  the  signs  of  the  terms  of  the  numera- 
tor of  a  fraction,  preceded  by  a  minus  sign,  when  clearing  of  fractions '? 
Upon  what  principle  (give  four  different  explanations)  may  the  signs 
of  all  the  terms  of  an  equation  be  changed  ? 


Exercise  80. 

1.  The  second  digit  of  a  number  exceeds  the  first  by  3 ; 
and  if  the  number,  increased  by  36,  be  divided  by  the 
sum  of  its  digits,  the  quotient  is  10.     Find  the  number. 

Solution.     Let  x  —  the  digit  in  tens'  place. 

Then  a;  +  3  =  the  digit  in  units'  place, 

and  2  a;  +  3  =  the  sum  of  the  digits. 

Therefore,  10  a:  +  a:  +  3,  or  11  a:  +  3  =  the  number. 

lla:+3  +  36 
Hence,  — ^ — r-5 — =  10.     .-.  a:=  1.     lla:+  3  =14,  the  number. 


PROBLEMS.  211 

2.  The  first  digit  of  a  number  is  three  times  the  second ; 
and  if  the  number,  increased  by  3,  be  divided  by  the  differ- 
ence of  the  digits,  the  quotient  is  17.     Find  the  number. 

3.  The  first  digit  of  a  number  exceeds  the  second  by  4 ; 
and  if  the  number  be  divided  oy  the  sum  of  its  digits,  the 
quotient  is  7.     Find  the  number. 

4  The  second  digit  of  a  number  exceeds  the  first  by  3 ; 
and  if  the  number,  diminished  by  9,  be  divided  by  the 
sum  of  its  digits,  the  quotient  is  3.     Find  the  number. 

5.  A  can  do  a  piece  of  work  in  7  days,  and  B  can  do  it 
in  5  days.  How  long  will  it  take  A  and  B  together  to  do 
the  work  ? 

Solution.    Let  x  =  the  numler  of  days  it  will  take  A  and  B  to- 
gether. 

Then  -  =  the  part  they  do  in  one  day  ; 

but  =  =  the  part  A  can  do  in  one  day, 

and  e  =  the  part  B  can  do  in  one  day. 

Therefore,    =  +  ^  =  the  part  A  and  B  can  do  in  one  day. 
7      o 

Hence,  -  =  ^  +  ^.    Therefore,  x  =  2\^. 

6.  A  can  do  a  piece  of  work  in  2 J  days,  B  in  3  days, 
and  C  in  5  days.  In  what  time  will  they  do  it.  all  work- 
ing together  ? 

7.  A  can  do  a  piece  of  work  in  a  days,  B  in  6  days, 
C  in  c  days.  In  what  time  will  they  do  it,  all  working 
together  ? 


212  ELEMENTS  OF  ALGEBRA. 

8.  A  and  B  together  can  do  a  piece  of  work  in  12  days, 
A  and  C  in  15  days,  B  and  C  in  20  days.  In  what  time 
can  they  do  it,  all  working  together  ? 

9.  A  and  B  together  can  do  a  piece  of  work  in  a  days, 
A  and  C  in  6  days,  B  and  C  in  c  days.  In  what  time  can 
they  do  it,  all  working  together  ?  In  what  time  can  each 
do  it  alone  ? 

10.  A  tank  can  be  emptied  by  three  pipes  in  80  min- 
utes, 200  minutes,  and  5  hours,  respectively.  In  what 
time  will  it  be  emptied  if  all  three  are  running  together  ? 

11.  A  sets  out  and  travels  at  the  rate  of  9  miles  in  5 
hours.  Six  hours  afterwards,  B  sets  out  from  the  same 
place  and  travels  in  the  same  direction,  at  the  rate  of  11 
miles  in  6  hours.     In  how  many  hours  will  he  overtake  A  ? 

Solution.     Let  x  —  the  number  of  hours  B  travels. 

Then  x  +  6  =  the  numher  of  hours  A  travels; 

also,  I  =  the  numher  of  miles  per  hour  A  travels, 

and  i^-  =  the  number  of  miles  per  hour  B  travels. 

Then,  y^  x  =  the  number  of  miles  B  travels, 

and  I  (a:  +  6)  =r  the  number  of  miles  A  travels. 

Hence,  V"  ^  =  f  (^  +  6).     Therefore,  x  =  324. 

12.  A  man  walked  to  the  top  of  a  mountain  at  the  rate 
of  2  miles  an  hour,  and  down  the  same  way  at  the  rate  of 
3^  miles  an  hour,  and  is  out  13  hours.  How  far  is  it  to 
the  top  of  the  mountain  ? 

13.  A  person  has  a  hours  at  his  disposal.  How  far 
may  he  ride  in  a  coach  which  travels  b  miles  an  hour,  so 
as  to  return  home  in  time,  if  he  can  walk  at  the  rate  of  c 
miles  an  hour  ? 


PROBLEMS.  213 

14.  In  going  a  certain  distance,  a  train  travelling  55 
miles  an  hour  takes  3  hours  less  than  one  travelling  45 
miles  an  hour.    Find  the  distance. 


15.  The  distance  between  London  and  Edinburgh  is 
360  miles.  One  traveller  starts  from  London  and  travels 
at  the  rate  of  5  miles  an  hour ;  another  starts  at  the  same 
time  from  Edinburgh,  and  travels  at  the  rate  of  7  miles  an 
hour.     How  far  from  London  will  they  meet  ? 

16.  The  distance  between  A  and  B  is  154  miles.  One 
traveller  starts  from  A  and  travels  at  the  rate  of  3  miles 
in  2  hours ;  another  starts  at  the  same  time  from  B,  and 
travels  at  the  rate  of  5  miles  in  4  hours.  How  long  and 
how  far  did  each  travel  before  they  met  ? 

17.  The  distance  between  A  and  B  is  a  miles.  One 
traveller  starts  from  A  and  travels  at  the  rate  ot  711  miles 
in  n  hours ;  another  starts  at  the  same  time  from  B,  and 
travels  at  the  rate  of  b  miles  in  c  hours.  How  long  and 
how  far  did  each  travel  before  they  met? 

1 8.  If  it  takes  m  pieces  of  one  kind  of  money  to  make 
a  dollar,  and  ?i  pieces  of  another  kind  to  make  a  dollar, 
how  many  pieces  of  each  kind  will  it  take  to  make  one 
dollar  containing  c  pieces  ? 

19.  The  denominator  of  a  certain  fraction  exceeds  the 
numerator  by  6 ;  and  if  8  be  added  to  the  denominator, 
the  value  of  the  fraction  is  J.     Find  the  fraction. 


20.   A  can  do  a  piece  of  work  in  2  m  days,  B  and  A 
gether  in  n  days,  and  A  and  C  in  m  +  ^ 
time  will  they  do  it,  all  working  together  ? 


together  in  n  days,  and  A  and  C  in  m  +  ^  days.     In  what 


214  ELEMENTS  OF  ALGEBRA. 

21.  In  a  composition  of  a  certain  number  of  pounds  of 
gunpowder  the  nitre  was  10  pounds  more  than  ^  of  the 
whole,  the  sulphur  was  4^  pounds  less  than  J  of  the  whole, 
and  the  charcoal  2  pounds  less  than  ^  of  the  nitre.  Find 
the  number  of  pounds  in  the  gunpowder. 

22.  A  broker  invests  |  of  a  certain  sum  in  5  %  bonds, 
and  the  remainder  in  6  bonds;  his  annual  income  is 
$180.  Find  the  amount  in  each  kind  of  bond,  and  the 
sum. 

23.  A  broker  invests  —  th  of  a  certain  sum  in  a  %  bonds, 

n 

and  the  remainder  in  c  %  bonds ;  his  annual  income  is  b 
dollars.  Find  the  amount  in  each  kind  of  bond,  and  the 
sum  invested. 

24.  At  the  same  time  that  the  west-bound  train  going 
at  the  rate  of  33  miles  an  hour  passed  A,  the  east-bound 
train  going  at  the  rate  of  21  miles  an  hour  passed  B ;  they 
collided  18  miles  beyond  the  midway  station  from  A. 
How  far  is  A  from  B  ? 

25.  A  person  setting  out  on  a  journey  drove  at  the  rate 
of  a  miles  an  hour  to  the  nearest  railway  station,  distant  h 
miles  from  his  home.  On  arriving  at  the  station  he  found 
that  the  train  had  left  c  hours  before.  At  what  rate  should 
he  have  driven  in  order  to  reach  the  station  just  in  time 
for  the  train  ? 

26.  A  merchant  drew  every  year,  upon  the  money  he 
had  in  business,  the  sum  of  a  dollars  for  expenses.  His 
profits  each  year  were  the  nth.  part  of  what  remained  after 
this  deduction,  but  at  the  end  3  years  he  found  his  money 
exhausted.     How  many  dollars  had  he  in  the  beginning  ? 


SIMULTANEOUS   SIMPLE  EQUATIONS.  215 

CHAPTER  XVL 

SIMULTANEOUS  SIMPLE  EQUATIONS. 

88.  Simultaneous  Equations  are  such  as  are  satisfied  by 
the  same  values  of  the  unknown  numbers. 

Thus,  3 X  +  y  =  9  and  5a:  —  2y  =  4  are  satisfied  only  hy  x  =  2 
and  y  =  S. 

Elimination  is  the  process  of  combining  simultaneous 
equations  so  as  to  cause  one  or  more  of  the  unknown 
numbers  to  disappear. 

This  process  enables  us  to  fonn  an  equation  containing  but  one 
unknown  number.  The  equation  thus  formed  can  be  solved  as 
shown  in  the  preceding  chapter. 

Hote.    There  are  only  three  methods  of  elimination  most  commonly  used. 

Elimination  by  Addition  or  SnbtractioiL 

89.  Example  1.     Solve  the  equations  :   5  3a; -5?/ =13        (1) 

^  l2x  +  7y  =  S\        (2) 

Hote  1.  The  abbreviations  (1),  (2),  (3),  etc.,  read  "equation  one,"  "equa- 
tion two,"  etc.,  are  used  for  convenience  to  distinguish  one  equation  from 
another. 

Solution.  To  eliminate  x  we  must  make  its  coefl&cients  equal  in 
both  equations.  Multiplying  the  members  of  (1)  by  2,  and  those 
of  (2)  by  3,  we  have 

5  6  X  -  10  y  =    26         (3) 
i6a;  +  21y  =  243        (4) 


216  ELEMENTS  OF  ALGEBRA. 

Subtracting  the  members  of  (3)  from  the  correspojiding  members 
of  (4),  we  have  31  y  =  217.  .'.y  =  1.  Substituting  this  value  of  y 
in  (1),  we  obtain  3  a;  -  35  =  13.     .-.  x=  16. 

VerifiGation.     Substituting  16  for  x,  and  7  for  ^  in  (1)  and  (2), 


we  have   548-35  =  13      (1), 
we  nave   "[gg,  49^31      (2), 


,  ,    identities. 
32  +  49  =  81      (2), 

Votes :  2.  In  this  sohition  we  eliminate  x  by  subtraction.  But  suppose  we 
wish  to  eliminate  y  instead  of  x.  Multiply  (1)  by  7,  and  (2)  by  5,  then  add 
the  resulting  equations,  and  we  have  31  ic  =  496.  .  •.  ic  =  16.  This  value  of  x 
substituted  in  (1)  gives  y  =  1. 

3.  When  one  of  the  unknown  numbers  has  been  found,  we  may  use  any  one 
of  the  equations  to  complete  the  solution,  but  it  is  more  convenient  to  use  the 
one  in  which  the  number  is  less  involved. 

4.  It  is  usually  convenient  to  eliminate  the  unknown  number  which  has  the 
smaller  coefficients  in  the  two  equations.  If  the  coefficients  are  prime  to  each 
other,  take  each  one  as  the  multiplier  of  the  other  equation.  If  they  are  not 
prime,  find  their  L.  C.  M.,  divide  their  L.  C.  M.  by  the  coefficient  in  each  equa- 
tion, and  the  quotient  will  be  the  smallest  multiplier  for  that  equation. 

Example  2.     Solve  the  equations  :   515^  +  ^7^  =  92  (1) 

^  (  55  a:  -  33  7/  =  22  (2) 

Solution.  Multiplying  the  members  of  (1)  by  11  (the  quotient 
of  165  divided  by  15),  and  those  of  (2)  by  3,  we  have 

5  165  a;  +  847  y  =  1012  (3) 

Xl^bx-    99  2/=      66  (4) 

Subtract  the  members  of  (4)  fron  the  corresponding  members  of 
(3),  9461/ =  946.  .-.  ?/=  1.  Substitute  this  value  of  y  in  (1), 
15ar+77  =  92.     .-.  a;  =  1. 

Proof.     Substituting  1  for  x,  and  1  for  ?/  in  (1)  and  (2),  we  have 

5  15  +  77  =  92         (1) 
1  55  -  33  =  22         (2) 

Hence,  both  equations  are  satisfied  for  a:  =  1  and  2/  =  !• 

Example  3.     Solve  the  equations  :    5  ^7  a:  -  12 !/  =  289  (1) 

^  (  55  a;  +  27  2/  =  491  (2) 


SIMULTANEOUS  SIMPLE  EQUATIONS.  217 

Process.     Multiply  (1)  by  9,    693  x  -  108  y  =  2601  (3) 

Multiply  (2)  by  4,                        220  x+lOSy=  1964  (4) 

Add  (3)  aiid  (4),                          913  x                =  4565.  .-.  x  =  5. 

Substitute  this  value  of  x  in  (2),     275  +  27  y  =    491.  .♦.  y  =  8. 

Prool    Substitute  5  for  x,  and  8  for  y  in  (1)  and  (2),  and  we 
have   J  279  =  279     (1),     .^^^^^ 
\  491  =  491      (2), 
Let  the  student  supply  the  method  from  the  solutions. 


Exercise  81. 

Solve  the  following  simultaueous  simple  equations 


1. 

|3a;  +  4y=10. 
Ux"+    y=    9. 

8. 

(Jy  +  Ja;  =  26.» 
(fy  +  |.;  =  25. 

2. 

Sx-     y  =  34. 
a;  +  8  y  =  53. 

9. 

( .25  x  +  4.5y  =  10. 
1. 75  y-. 15  a;  =  .9. 

3. 
4 

10  a:  +    9y  =  290. 
12  2;-lly  =  130. 

7  y  -  3  a;  =  139. 
2x  +  5ij=    91. 

10. 

J  3^  2       '' 
l2  +  3  =  ^- 

5. 

{6x-5y  =  -7. 
\  10  a;  4- 3  7/ =  11. 

11. 

(  .5  a:  +   2y=  1.8. 
1  .5  y  -  .8  a:  =    .08. 

6. 

9a;-4y  =  -4. 
15  aj  +  8  y  =  -  3. 

12. 

(7a:+^y  =  99. 
1  7  y  4-  j  a:  =  51. 

7. 

9  y  4-2  a;  =15. 
4y4-7a;=    3. 

13. 

r    Jaj4-  3y  =  22. 
1  l\x-\y=20. 

♦  Clear  of  fractions  first. 


218  ELEMENTS  OF  ALGEBRA. 


Elimination  by  Substitution. 

90.    Example.    Solve  the  equations :    HaJ  +  32/  =  22  (1) 

^  l5x-7y=    6  (2) 

Solution.     From   (2),    x  =  — — -^     (3).     Since  the  equations 

o 

are  simultaneous,  x  means  the  same  thing  in  both,  the  substitution 
of  this  value  of  x  in  (1),  will  not  destroy  the  equality.      Hence, 

4/ — F~^)  +3^  =  22.      Clearing  of  fractions,   transposing,   and 

uniting  like  terms,  43  2^  =  86.     .'.  y  =  2.    Substitute  this  value  of 
y  in  (3),  x  =  4. 

Let  the  student  supply  the  method. 


Exercise  82. 

Solve  by  substitution : 

1 


2. 


3. 


x+Sij^U.  ^-    l^x  +  iy  =  7. 

7  x  +  4:y  =  29.  {S7J  +  4cX  =  SS. 

Sx+     ?/  =  ll.  \5x+62j=61. 

l^V-^x  =  21.  l3+2  =  l- 


I  .08  2/  -  .21  a  =    .33.  (  3  y  -  4  a;  =  1. 

I    .7z  +  .12y=  3.54  I  3 a;  -  2 ?/  =  1. 

"    ~  *•   10.  I  11      -^ 

^--  =  0. 


^  =  1. 


SIMULTANEOUS  SIMPLE  EQUATIONS.  219 

Ml2/-7^  =  37.  (10a:  =  9  +  7y. 

^^-     |8y  +  9a:  =  41.  U2/  =  15a!-7. 

12.    <  aud  verify. 

Elimination  by  Comparison. 

91.    This  method  depends  upon  the  following  axiom  : 

6.    Things  equal   to  the  same   thing  are  equal  to  each 
other. 

Example.    Solve  the  equation  .   )^^-^y=^  0) 

^  (7x-4y  =  8^  (2) 

Solution.     From  (1),  x  =  i±A^     (3).     From  (2),  x  =  ?i±l^. 

Since  these  equations  are  simultaneous,  x  means  the  same  thing 

in  both,  — ^  =  -^ ^ .     Solving  for  y,  we  have  y  =  4.     Sub- 

7  1  +  20 

stituting  this  value  in  (3),  x  =  — tj —  =  3^. 

Let  the  student  supply  the  method. 

Exercise  83. 
Solve  by  comparison : 

5a;+6y  =  — 8.  j6x+l5y  =  —  6. 

3x  +  4y  =  -5. 


(6x+  l5y  =  -{ 
I  8  a;  -  21  y  =  74 

\}x-{-iy  =  S. 

(-^x  +  3y  =  51.  rSy    -.7x    =.4. 

'  \7x+2y  =  3.  I  .02  y  +  .05  a:  =  .2$ 


12^-7^=17. 
•   U2:+8y  =  20.  ^ 


220  ELEMENTS   OF  ALGEBRA. 


(l.lx    -l.Sy    =0. 
^-  t.l3:r-    .11 7/ =  .48.  \l  + 


0^  =  3  2/ -23.  ^^1      5  2      ~'^* 


12.  -{  ^ 


1  =  42. 


r.30^-.772/=:-2.95.  h^  4.y^4o 

1 .20 a: +.21^=1.65.  ^8"^  9 


92.  Each  of  the  equations  should  be  reduced  to  its  simplest  form, 
if  necessary,  before  applying  either  method  of  elimination. 

Notes :  1.  An  expeditious  method,  for  the  solution  o^ particular  examples, 
is  that  of  first  adding  the  given  equations,  or  subtracting  one  from  the  other. 

2.  Usually,  in  solving  examples  of  two  unknown  numbers,  it  is  expedient 
to  find  the  value  of  the  second  by  substitution;  but  this  is  by  no  means 
always  so. 

Example.     Solve : 

2y  +  42:-2tf  10|i/-5fx-18 

3a:+.y       13?/-37^_  9-9a;-t/       10a:+.25i/-10.5 

L~T2^+        44        -^^+        22~^~  33  ^^^ 

Process.     From  (1),  127  y+    59  x=  1928  (3) 

From  (2),  59  ^  +  127  rr  =  1792  (4) 

Adding  (3)  and  (4),  186  y  +  186  a;  =  3720  (5) 

Dividing  (5)  by  186,  y+         a;  =      20  (6) 

Subtracting  (4)  from  (3),  68?/-    68  a;  =    126  (7) 

Dividing  (7)  by  68,  y  -         x  =        '2.  (8) 

Adding  (6)  and  (8),  %y                =22.  .-.2^=11. 

Subtracting  (8)  from  (6),  2x~      18.  .-.  a:=    9. 


Solve 


SIMULTANEOUS   SIMPLE   EQUATIONS.  221 

Exercise  84. 


1     fy(^  +  7)  =  a:(y+  1).  (2y  +  Ax    =1.2. 

I  2y+20  =  32;+  1.  XsAy -.02x=    .01. 

r(y+l)(2:+2)-(y  +  2)(2:+l)  =  -l. 
^-  \  3  (y  +  3)  -  4  (2:  +  4j  =  -  8. 

f  .3  2:  +  .125y  =  a:- G.  rx-4y  =  -3. 

l3a:-.5y  =  28~.25y.  I  aj  +  v     =32. 


.5y  =  28~.25y.  {x  +  y 

6.  -^ 


'4:X  +  Sy      2  y -\-7--x  _         X--S 
To  24  -^"^      5 


9a;+52/  — 8      a;  +  y  _  7y  4-  6 


12 

4 


9.  ^ 


<^  5  +  y       12  + a; 
l2a;+  53^  =  35. 


10.  -^ 


3y-10(a:~l)      ^j-y  .   .  _  ^ 
6  -^      4      "^  ^  -  "• 

'4a;-3y-7  _  3_^  _  2  .y  _  5 
5  "  10        15       6  * 

y~l  ,  ?  _  3y  _         7/^^  4.  ?  4.  JL 
L    3      "^2       20  15     "^  6  ■*■  10 


-3  5     ~       4       *  U"^il       33 


222  ELEMENTS  OF  ALGEBRA. 

r:.  +  l(3^.-y-l)^i+f(2/-l). 

f2x  _Sy  —  2  _     _  4:  +  X      y  —  x 


14.  ^  18 

by 


\2x. 

2Zy  -X 


2x-^  = 


a?  +  43      X  —  y 
2.4  X  +  .32  y 


\  X  -]r  "i^o      X  —  y 

.36  X  -  .05 


\'. 

1-3.V 

7 

=  2i. 

3^/  + 
11 

-^-9  = 

=  —  ii;. 

«  ^  _L 

2.6  + 

.005  3/ 

17. 


.5  '  .25 

04  2/ +  .1       .07  a? -.1 


18. 


.6 


_  Zx-2-\-,y  ^  IBy  +  jx 

^  11  "^33 


2x-h3y      x-5       11 :?/  +  152       3  a;  +  1 


r 


?/  -  2       10  -  y      ^  -  10 


19  ^      ^  ^  ^ 

■^22;  +  4       0^  +  4?/ +12 


I 


8 


21    H(2^+ 72/)- 1=1(20.-62/+!). 

I  a?  =:  4  2/. 


SIMULTANEOUS  SIMPLE  EQUATIONS.  223 

Suggestion.  Multiply  the  members  of  the  first  equation  by  2, 
transpose,  and  unite  like  teims  ;  then  clear  the  resulting  equation  of 
fractions.  Multiply  the  second  equation  by  3,  transpose,  and  unite 
like  terms;  etc. 


23.  < 


24.  < 


25. 


r2         ,      y  ^      3?/       1 

,1-1  +  2  =  1-2.. 6. 

f6x  +  9       3x-5  _  3x  +  4: 

4       +4y~6~    i"^        2 
8a;  4-  7      Sx-  6y_         9-4a; 
^10  2a:-8~  5 

16  +  60x  _16xy-  107 
3y-l    ~      5  +  2y      • 


Suggestion.      Multiply  the  members  of  the  first  equation  by 
5  +  2  y,  transpose,  and  unite  like  terms  ;  then  clear  of  fractions  ;  etc. 


26. 


27. 


X  —  y  __1 
X  -\-  y      5  * 

13  3 


y+2a;+3  4?/  —  5a;+6 

3  ^19 

6y-5a;  +  4~  3y  +  2a;+  l' 

ry-x  =  l.  r5(y+3)  =  3(ic~2)  +  2. 

28.  I  y+  1  _  y-1  _  6      29.  ^      2     ^     3 
U— 1  X      ~  7'  i^y+3~a:  —  2* 


224  ELEMENTS  OF  ALGEBRA. 


30. 


31. 


32.  < 


i(2  2/  + 7^) -1  =  1(2  2/- 6^+1). 


U  = 


y 


6y2-24a^+130 
2y-4:x+  3 
151  -  16y_  9  0^3/ -110 
4^-1  3a;-4     ' 

^4a;  +  22/       4a;+53/ 
""16  31        ""    • 

2^+j/      3  ?/  -  2  0?      36 

~5     +""~6     =y 

r5.T  +  202/  =  .l.  04  J  2/-^      3-"- 

^''-  \ll^  +  302/  =  -.9.  l^+_^±i_7^Q 

V2/  —  a;  —  1 


35. 


r        2a;— .5y     5jy— 19a;— 15       „  ^— a;  +  2 


93.  Fractional  simultaneous  equations  in  which  the  unknown 
numbers  occur  in  the  denominators  as  simple  or  like  expressions^  are 
readily  solved  without  previously  clearing  of  fractions.     Thus, 


Example  1.    Solve: 


^'h 

21 

y 

= 

10 

20 

6 

X 

y 

- 

2 

(1) 

(2) 


Solution. 

10      3 
Dividing  the  members  of  (2)  by  2,  we  have  —  —  -  =  1   (3).     Mul- 

70      21 
tiplying  the  members  of  (3)  by  7,  —  —  -—  =  7     (4).     Adding  the 

X        y 


SIMULTANEOUS  SIMPLE  EQUATIONS.  225 

members  of  (1)  and  the  corresponding  members  of  (4),  we  have 

.  X  =  5.     Substituting  this 


85  5 

—  =  17.      Dividing  by   17,  -  =  1 

X  X 


value  of  X  in  (1),  gives  -  =  1:     ..  y  =  3. 

Note-  If  we  cleared  these  equations  of  fractions  they  would  give  the  pro- 
•  liict  xy,  and  thus  become  quite  complex.  In  the  solution  of  this  particular 
.lass  of  examples  it  is  always  easier  to  eliminate  one  of  the  xmknown  numbers 
without  clearing  of  fractions. 


Example  2.    Solve: 


2-y^rx  =  ^ 
2         4   _ 


2         20        136 
Process.  Multiply  (0  by  i,  3T  +  g^-  =  ^ 


Subtract  (3)  from  (2), 
Simplify  (4), 


y 

4_ 
5x 


2,1  x 
20 
■  27  X 

8 
135  X 


208 

9 
208 

9  • 


2  2 

Substitute  in  (2),  ^  -  312  =  -8,  or  «    =  304. 

^y  ^y 


(1) 

(2) 

(3) 
(4) 


^~      390' 


y  = 


456 


Exercise  86. 


Solve: 


1.  < 


2. 


r2    1    ,^ 
-  +  -  =  10. 

X       y 


9  y       2x~      ^' 
^3y      4  a;       6  * 


226 


ELEMENTS  OF  ALGEBRA. 


5_ 
12 

24 


12. 


?-^  =  16. 

2/       2a; 

14,^ 
+  -  =  _  15. 

l2y      X 


7. 


y  +  l 


__7^ 
~  12' 

12* 

_  5 

~  6' 

=  2. 


13. 


14. 


2  2/       4a? 


13^/^22; 


71 
'1- 


79. 


15.  < 


5       16 

-  +  — 
X        y 

=  44. 

y^x        y 


9. 


ri5 

8 

17 

2/ 

X  ~ 

3 

2 

3 

7 

I  y 

a; 

5 

16. 


11 

2^ 

+ 

6 
32/ 

17. 

17 

6  1/ 

— 

5 
a; 

3 
2* 

10. 


1 


+ 


2  (2;  -  2)    '   3  (2  2/ 
3  5 


1) 


5  a; -10       4  (4  2/ -2) 


=  5. 


=  1. 


11. 


22/  2 


17. 


2a; 


2 

5 

4 

"27 

1 

42/ 

1 

11 

"72 

18. 


SIMULTANEOUS  SIMPLE  EQUATIONS.  227 

a;  —  2      y  +  2  ^g 

3  1  1 


^a;  — 2       y  +•  2       2 


1       1        7 
2x     3y""15* 
a:  —  5  v/  +  4  a:  y 

1 

4  a;  V 

15 

19.   Suggestion.    Reduce  the  first  member  of  the  second  equa- 
tion to  mixed  expressions.     Etc. 

('2i  _  3y  ^  2  rL2.  25^86 

l3^      3y-2a;+l  3'2lJ^2^ 
5,            2y           _  *    |25_1_6^ 

a;"^3y-2a;+l  la;       y 


94.  In  solving  literal  simultaneous  equations,  either  of  the  pre- 
ceding methods  of  elimination  may  be  applied,  usually  the  method 
by  addition  or  subtraction  is  to  be  preferred. 

Note.  Numbers  occupying  like  relations  in  the  same  problem,  are  generally 
represented  by  the  same  letter  distinguished  by  different  subscript  figures  ;  as, 
«l  ;  «2 ;  "8  >  fitc*  ?  r«*d  a  one  ;   a  ttoo  ;  a  three  ;  etc. 

They  may  also  be  represented  by  different  euxents ;  as,  a';  a";  a'";  etc.; 
read  a  prime;  a  second;  a  third;  etc. 


Kt AMPLE  1.     Solve:  1^  ^ 

+  n 

y 

=  a 

(1) 

-hn, 

iV 

=  «i 

(2) 

Process.     Multiply  (1)  by 

mj. 

m^mx  +  m^ny  =  m^a 

(3) 

Multiply  (2)  by  m. 

m^mx  +  mn^y  =  ma 

(4) 

Subtract  (4)  from  (3), 

m^ny  —  mn^y  =  mjO  - 

-w»«i, 

or  factoring, 

(m^n  —  mn^)y  —  m^a  - 

-  may 

Dividing  by  mjn  —  muj, 

^""h 

-  Oj  m 

^      nij^n  - 

-mnj 

Multiply  (1)  by  rij. 

mn^x  -f-  nn^y  =  n^a 

(6) 

Multiply  (2)  by  n, 

m^ux  -\-  nn^y  =  na^ 

(6) 

Subtract  (6)  from  (5), 

mn^x  —  m^ux  =  71^0  - 

-no,. 

or  factoring, 

(mui  —  in^n)x  =  n^a  - 

-  noj. 

Therefore, 

-_a^n 
-m^n' 

228 


ELEMENTS  OF  ALGEBRA. 


Example  2.    Solve:   ^ 


f     X  y 

1^+5  +  ^='" 


x  + 


(1) 

(2) 


[  2a6  ~  a2  +  62 

Proceas.     Free  (2)  from  fractions,  transpose,  and  factor, 

{a-hyx-{a^-hyy^Q  (3) 

Simplify  (1),  {a-h)x^{a^lS)y  =  2a(a+b)  (a-b)       (4) 

Multiply  (4)  by  a -ft,   (a-byx+(a^-b'^)y  =  2a(a+b)  {a-by    (5) 
Subtract  (3)  from  (5),  2  a  (a  +  6)  i/  =  2  a  (a  +  6)  (a  -  &)2. 

Divide  by  2 a  (a  +  6),  y  =  (a-b)\ 


Substitute  in  (1), 


Examples.   Solve 


a+b 


+  a-b  =  2a.     .'.x=(a  +  by. 


f       m  n—'m(m+n)(b—y)__ 

J  n{a-\-x)  m(b-~y)         ~ 

!  ryt 


I  — —  +  T =  m2+n2 

\^a  +  x      b—y 


Process.     From  (1), 
Multiply  (3)  by  ~, 


+ 


n(a  +  x)       m{b  —  y) 


=  m  +  n 


+ 


a  +  X      m^(b  —  y)      m 


—  (m4-n) 


(2) 

(3) 
(4) 


Subtract  (4)  from  (2),        ^3^  -  ^2Q,_y^ 

1 


Simplifying, 
Substitute 


m{b  —  y)  ^  m 


m(b-y) 


=  1  or 


b-y 

n 
a  +  x 


m2  in  (2), 


=  n^    .-.  X 


Example  4.    Solve:   < 


x-y  +\ 
x-y-1 

x  +  y  +  I 


x  +  y 


-  a  =  0 
-6  =  0 


(1) 
(2) 


SIMULTANEOUS  SIMPLE  EQUATIONS.  229 

Procesa.     From  (1),       {a-  l)x -  (a-  l)y  =  a  +  I  (3) 

From  (2),  (6- 1):.  +  (6- l)y  =  6  + 1  (4) 

Divide  (3)  by  a  - 1,  x~y  =  ^—^^  (5) 

Divide  (4)  by  6  - 1,  x  +  y  =  ^-3-j  (6) 

2(a6-l) 
Add  (5)  and  (6),  2 x  =  (q_i)(fc_i^  ' 

ab-l 
•••^-(a-l)(6-l)- 

2(a-6) 
Subtract  (5)  from  (6),  2 y  =  ^^.^^^^.^^  • 

a-b 


Exercise  86. 

Solve : 

-     (ax-\-hy  =  m.  ^    (  ax  -\-  by  =  a^, 
'\bx-\-ay  =  n.  '  \hx  -\-  ay  =  1?. 

^    nx  +  my  =  n.  ^     (px'-qy  =  r, 
'  \px-\-qy  =  r,  '  \rx—py  =  q, 

^    (ax  =  hy.  ^(x  +  ay  =  ai. 
'  \bx-\-ay  =  c,  '  \ax-\-aiy=l. 

-?  +  ?=!.  (^  +  ^-  =  a. 

.    J  a       b      ab  ^J^     \  m       n 

I      bi      aibi  ^n  '  m 


5.  < 


3j/^2^^2  r_y X    ^    1 

m         ?i  *  -^   Ja-{-b      a  —  b      a  +  b 

9y_6^^3  '  I     ?/     ,     ^    ^    ^ 


230  ELEMENTS  OF  ALGEBRA. 

=  (a  +  h)  y. 


'  \  cy  +  hx  =  a.  '  \x  +  y  =  c. 


rih;  /     J    14.    -  +  2'  =  2. 

l-^  +  r  =  cll+-).  {mx  =  ny 


\^(m  —  n)  y  =  {m  -{•  n)  X. 


y      X       ^ 
a       ai 


(ax-hy 

\(a-h)x+  (a  +  b)y=2  {a^  -  l^). 

22     (m{m-7j)  =  n(x  +  y-m). 
\m  (x  —  n  —  y)  =  n  (x  —  n). 

a  b  a  fx  +  y+l^m  +  l 


2g^a  +  a?       b-y       b         24.  <^2/-^+l       ^"1 

b  a  b  \  X  +  y  +  1  __  n+1 

a  +  x       b  —  y~a  \y  —  x-^l~l—n 

25    /  3/  -  ^  +  2  (m  -  ?i)  =  0. 

■  1  (a:;  +  7i)  (y  +  m)  —  (y  —  m)  (x  —  n)  =  2  (m  —  n)\ 


SIMULTANEOUS  SIMPLE  EQUATIONS. 


231 


95.  Simultaneous  equations  with  three  or  more  unknown  num- 
bers are  solved  by  eliminating  one  of  the  unknown  numbers  from  the 
given  equations  ;  then  a  second  from  the  resulting  equations  ;  and  so 
on,  until  finally  there  is  but  one  equation  with  one  unknown  niunber. 
Thus, 

r  2  y  +  2  +  2  y  =  -  23  (1) 

y-f-32  =  -    2  (2) 

4a:  +  z=13  (3) 


Example  1.    Solve 


3  +  3. 


Process.     Multiply  (2)  by  2, 
Subtract  (5)  from  (1), 
Multiply  (4)  by  12, 
Subtract  (7)  from  (3), 
Multiply  (8)  by  5, 
Add  (9)  and  (6), 
Substitute  in  (4), 
Substitute  in  (3), 
Substitute  in  (2), 


-  20  (4) 

2y  +  6z  =  -     4  (5) 

-52  +  2y  =  -    19  (6) 

4a; +  361;  =  -240  (7) 

z-36v=    253  (8) 

52- 180i;=  1265  (9) 

-178y=  1246.  .-.  r  =  -  7. 

I -21  =-20.  .-.  x  =  2. 

12  +  2=13.  .-.    2=1. 

y  +  3  =  -2.  .-.   y  =  -5. 


Proof.     Substituting  —  7  for  y,  3  for  x 

f  -  23  =  - 

-  2  =  - 
13=13 

-  20  =  -  : 


(I),  (2),  (3),  and  (4),  we  have  -{ 


—  5  for  y,  and  1  for  z  in 
23     (1), 

^    ^^]'  identities. 
(3). 

(4), 


Kote.  When  the  values  of  several  unknown  numbers  are  to  be  found,  it  is 
necessary  to  have  as  many  simultaneous  equations  as  there  are  unknown 
numbers. 


EiLAMPLE  2.     Solve: 


J_      J 1__  1 

2z"^  4y      32~  4 


1 

1        4 


0) 

(2) 
(3) 


\2                          ELEMENTS  OF  ALGEBRA. 

Process.     Multiply  (1)  by  2,    -  -f 

1          2        1 
2y      3z-2 

(4) 

Subtract  (2)  from  (4), 

5          2        1 
6y~  3z~2 

(5) 

Subtract  (2)  from  (3), 

2         4 
I5y'^  z~^^ 

(6) 

Multiply  (5)  by  6, 

5      4 

(7) 

Add  (6)  and  (7), 

77       77 
16  y~  15- 

.'.  y  = 

:1. 

Substitute  in  (2), 

1       1 

.'.    X- 

:3. 

Substitute  in  (5), 

fl      1 

5  2        1 

6  3z~  2' 

.'.    z  - 

:2. 
(1) 

Example  3.    Solve  :    - 

1      1 

ly  +  i-" 

Process. 

s 

(2) 
(3) 

Add  (1),  (2),  and  (3),  ?+?+^  =  a  +  &  +  c  (4) 

Divide  (4)  by  2,  ^  +  J  +  1  =  ^±A±f  (5) 

Subtract  (3)  from  (5),  ^  =  ^±|-— 

Subtract  (2)  from  (5),  ^      «  +  c  -  6 


Subtract  (1)  from  (5), 


y  2 

1       &4-C  — a 


2 

«+&-c 

2 

y  — 

a-6  +  c 

2 

h-\rC  —  a 


SIMULTANEOUS  SIMPLE  EQUATIONS. 


233 


Exercise  87 

Solve: 

f'^x-    y+    z=    9. 
I.  ^  a:_2y+32=  14. 


r4:x-3y+2z  =  40. 

^<5x-i-9y-7z  =  4:7. 

Ua;+8y-32  =  97. 

r2x-32j+oz=  15. 

3.  <  32:+27/-    z=    8. 

V— a;+  5y  +  2s  =  21. 

rSx-Sy+     z=    0. 

4.<2a;-7y  +  42:=    0. 

v9  2:+5  7/+32=  28. 

rx  -\-  y  -\-  z=    5. 
6.  ^  3  7/-5x  +  72  =  75. 
19  y- 11  2+ 10=    0. 

r.65//- .95a:  =  .5. 

6.  <  5.1  a: -3.3  3  =  6. 

V20.3  2- 23.1a:  =  21. 

rax  •\-  hy  '\-  cz=iZ. 

7.  <  rt  x  —  6  y  +  c  2  =  1. 

Vaa:  +  6y  — •  C2  =  1. 

r.2a;  +  .ly  +  .32=14. 

8-  <  .52:+.4y+.a2  =  32. 

^.7y-.8a:+.9c=  18. 


9.  < 


2^  +  2  +  3  = 

-+2+1= 
a;      ?/ 


X      y      z 


^  +  -  +  ' 


6. 

--1, 

17. 

=  1. 
=  1. 
=  1. 


11.  < 


234 


ELEMENTS  OF  ALGEBRA. 


fv-hx  +  y  +  z  =  14:. 
\2v  +  x  =  2y  +  z-2. 
14^^  ^  Sv  -  X  +  2  y  +  2  z  =  19. 
\  V       X       y       z 


^a  —  X       h  —  y       c  —  z 

+ + =  0. 


X 


y 


15.  < 


a  —  X       h       c  _ 
X  y      z~ 

X        y       z 


0. 


15.   Suggestion.    Reduce  fractions  to  mixed  expressions.    Etc. 


'x-\-2y  =  9, 
16.  ^3  7/ +  4^  =14. 

72  +  V  =  5. 

^2v  +  52:==  8. 
rx  +  y=  1. 
\x  -\-  z  =  b. 


2       1_  3 

X       y       z 

18.  <  ^  -  -  =  2. 

z       y 

1       1_4 

x^  z~  ^' 


f4:y-\-Sx  +  z     2x  +  2z-y-\-l  _         y-z-5 


10 


15 


19.  < 


9?/  +  52:-2;s      2y  +  a;  — 3^_  7x  +  z+S     1 

r2                  ^      4          ~         11        ^  6  * 
5^  +  32      2y+Sx-z     ^               ^     Sy  +  2x+7 
—4 12 +  2. =  0.-1+ g 


Queries.     Upon  what  principle  is  elimination  by  addition  and 
subtraction  performed  ?     What  substitution  ?    What  comparison  ? 


SIMULTANEOUS   SIMPLE  EQUATIONS. 


235 


Miscellaneous  Exercise  88. 
Solve : 

'x+1      x-l       6  r4a;+y  =  ll. 


23 


y 


a(x^-y)  +  h{x-y)  =  l. 
&  (x  +  y)  =  1. 


7. 


4. 


5. 


10. 


X  —  a 


=  0. 


a  b 


0. 


'a;  +  2y=2-32-4i;. 

Sy  +  2x  =  S  —  4:z-5v. 

9v-82-3  =  -6a;--7y. 
^v  =  25  -4^-  16y- 64a:. 

■(m2  —  n2)  (5  ^^  4.  3  y)  =  (4  ^  _  n)  2  m  n. 

„       a  m  vF 
m^y 


3  a;        15 

/M- 

lf-!=»- 

Ihh' 

y    ^ 

12-3- 

.|!+l-^ 

1=^ 
,ai      bi ' 

-{-{m-\-n-\'a)nx  =  n^y-\-{m-\-2n)mn. 


11. 


3  V  +  a:  +  2  y  -  2  =  22. 
4a;-     y4-32  =  35. 
4v  +  3a;-2y  =  19. 
.21^  +  4^+2^  =  46. 


(-15  a;  =  24  2-  10  y  +  41. 

12.  \  15y=12a;- I62+  10. 

ll8y-(7  2-13)-=14a;. 


'Ul=z. 


13.  < 


X 

1 

a;   ■    z 


y 
2 
+  -  =  11. 


^U-3. 
y       2 


236 


ELEMENTS  OF  ALGEBRA. 


14. 


'x  +  y  +  z  +  v  =  14:. 
2x+Sy  +  4:Z+5v  =  54:. 
4:x  —  5y  —  7z+9v  =  10. 
:Sx  +  4:y  +  2z-3v  =  ll. 


16. 


)  X  +  Z  +  V  = 


=    5. 
10. 

X  +  y+  V  =    6. 
x  +  y  +  z=12. 


15. 


18. 


ax  +  by  =  2m. 
ax  -\-  cz  =  2  71. 
,h  y  +  c  z  =  6p, 


rmx  +  ny  +  pz  =  m. 

17.  <  mx  —  ny—pz  =  n, 

\mx  +py  +  nz  =  p. 


Sx 


2y      ,      \ly 
— +  1+      ^ 


10 


45 


4.X-2 


8  7 

55^+  71  y  +  1 
18 


4  a;  —  3  V  +  5       45  ■ 

-  +  — 7 


^lx  = 
v^2Z 


17  +  2z-Stc. 


20. 


2(0  +  22/). 
4. 

y=  2.25-\-.75u—5v. 
z  =  ll-^u. 


19.  {u=ly^  I X 


(  ax  + 
\ay  + 


b  X 
by 


cy  =  m. 

cx  =  n. 


X      y       m 

21.  <  -+  -  =  -. 
\x       z       jp 

I  1       1_  1 

\^z       y       n 


22. 


f  :^+2/+2;  =  a+J+c. 
i  a+x  =  b-\-y  =  c+z. 


9S  11-7^     2(5-lly)  ^  17.5  +  5y     312.5-360a; 
?j-x'^  ll{y-l)~      3-2/  36(a;  +  5)     ~ 


^3        4        1        ^^  fx      ^       ^ 

--^— +  -=    7.6.  -+l=4a;. 

X       by       z 

4 


_        .,-  +  -  =  16.1. 

^o  X     2y     z 


+  l  =  2y. 


SIMULTANEOUS  SIMPLE  EQUATIONS. 


237 


1 


y  + 


1 


y~ 


26.  < 


a;  — 


X — 


i-i 


y 
l-x  =  0. 


x-\-y  =  2m^ 


27. 


711 -{-li- 


mn 


m+n 


m—n-\- 


mn 
m—n 


28. 


m  n  V  ^ 
-  +  -  +  -  =  3. 
X       y       z 

m       n      P  _  ^ 
X       y"^  z~    ' 


2  m 

X 


n 


0. 


29. 


30.  ^ 


xy 

x-{-  y 

xz 


=    70. 
=    84. 


X  ■{-  z 

-^  =  140. 

y  +  z 


31.^ 


xy 

x+  y 

yz 
y  +  z 

xz 
^x  -\-  z 


xy 

— ^^  =  m. 


32.  ^ 


'ax-\-  by  +  cz  =  0. 
a^x  +  ly^y  +  (^z  =  0. 
.a^x  +  h^y'i-c?^z  =  0. 

'a^z=:2. 


x  +  y 

xz 

x+  z 

yz    _ 

^y  +  « 

n. 


{ax  +  a^y  —  a^yVc 

2(3a;-2y)_^_   bz-y 


34 


35. 


a;  —  2g 
3a;  —  2i 


m  —  n  + 


Sz-7 

n 


2x  —  Sz 


m 


ly-^  = 


w 


m  +  71 

2  7l6 


7/1^  +  7?l  71  +  71^ 


238  ELEMENTS  OF  ALGEBRA. 


CHAPTER  XVII. 

PROBLEMS  LEADING  TO  SIMULTANEOUS  EQUATIONS. 

96.  The  solutions  of  the  following  problems  lead  to  simultaneous 
simple  equations  of  two  or  more  unknown  numbers.  In  the  solution 
of  such  problems  the  conditions  must  be  sufficient  to  give  just  as  many 
equations  as  there  are  unknown  numbers  to  be  determined. 

Exercise  89. 

1.  If  5  be  added  to  both  numerator  and  denominator 
of  a  fraction,  its  value  is  f ;  and  if  3  be  subtracted  from 
both  numerator  and  denominator,  its  value  is  ^.  Find  the 
fraction. 

Suggestion.  Let  x  =  the  numerator, 

and  y  —  the  denominator. 

ra;  +  5_  3 
— T~K  —  T' 

By  the  conditions,  •{  y  ^ 

I  a:-3_  1 

Solving  these  equations,  x  —  1,  y  =\\. 
Therefore,  the  fraction  is  ^y- 

2.  A  certain  fraction  becomes  equal  to  3  when  9  is 
added  to  its  numerator,  and  equal  to  2  when  2  is  sub- 
tracted from  its  denominator.     Find  the  fraction. 

3.  Find  two  fractions  with  numerators  5  and  3,  respec- 
tively, whose  sum  is  ||,  and  if  their  denominators  are 
interchanged  their  sum  is  f. 


PROBLEMS.  239 

4.  A  certain  fraction  becomes  equal  to  §  when  the 
denominator  is  increased  by  3,  and  equal  to  J  when  the 
numerator  is  diminished  by  3.     Find  the  fraction. 

5.  A  fraction  which  is  equal  to  |  is  increased  to  {|  when 
a  certain  number  is  added  to  both  its  numerator  and  de- 
nominator, and  is  \  when  3  more  than  the  same  number 
is  subtracted  from  each.     Find  the  fraction. 

6.  If  a  be  added  to  the  numerator  of  a  certain  fraction, 
its  value  is  a ;  and  if  a  be  added  to  its  denominator,  its 
value  is  ^  (a  —  1).     Find  the  fraction. 


7.  Find  two  numbers,  such  that  two  times  the  greater 
added  to  one  fifth  the  less  is  36 ;  three  times  the  greater 
subtracted  from  eight  times  the  less,  and  the  remainder 
divided  by  9,  the  quotient  is  7|. 

8.  Find  two  numbers,  such  that  if  the  first  be  increased 
by  a,  it  will  be  m  times  the  second,  and  if  the  second  be 
increased  by  6,  it  will  be  n  times  the  first. 

9.  Find  two  numbers,  such  that  if  to  J  of  the  sum  you 
add  18,  the  result  will  be  21 ;  and  if  from  |  their  differ- 
ence you  subtract  |,  the  remainder  is  3.65. 

10.  A  farmer  sold  to  one  person  25  bushels  of  corn  and 
52  bushels  of  oats  for  S 38.30  ;  to  another  person  42  bush- 
els of  com,  and  37  bushels  of  oats  for  $35.80.  Find  the 
number  of  dollars  per  bushel  received  for  each. 

11.  A  farmer  sold  a  bushels  of  corn  and  b  bushels  of 
oats  for  m  dollars  ;  also  at  the  same  time,  c  bushels  of  com 
and  d  bushels  of  oats  for  n  dollars.  Find  the  number  of 
dollars  per  bushel  received.     Apply  the  result  to  10. 


240  ELEMENTS  OF  ALGEBRA. 

12.  A  grocer  bought  a  certain  number  of  eggs,  part  at 
2  for  3  cents  and  the  rest  at  5  for  8  cents,  paying  $7.50 
for  the  whole.  He  sold  them  at  23f  cents  a  dozen,  and 
made  $2  by  the  transaction.  How  many  of  each  kind  did 
he  buy  ? 

13.  A  grocer  bought  a  certain  number  of  eggs,  part  at 
the  rate  of  a  eggs  for  m  cents  and  the  rest  at  the  rate  of  h 
eggs  for  n  cents,  and  paid  c  dollars  for  the  whole.  He 
sold  them  at  d  cents  a  dozen,  and  made  ^  dollars  by  the 
transaction.  How  many  of  each  kind  did  he  buy  ?  Apply 
the  result  to  12. 


14.  A  number  is  expressed  by  three  digits.  The  sum  of 
the  digits  is  8 ;  the  sum  of  the  first  and  second  exceeds 
the  third  by  4;  and  if  99  be  added  to  the  number,  the 
digit  in  the  units'  and  hundreds'  place  will  be  inter- 
changed,    rind  the  numbers. 

Suggestion.     Let  2  =  the  digit  in  units'  place, 

and  y  =  the  digit  in  tens'  place, 

also  X  —  the  digit  in  hundreds'  place. 

Hence,     100a:+  10y  +  2  =  the  number, 
and  100  2+  10^  +  a:  =  the  number  with  the  digit  in  units' 

and  hundreds'  place  interchanged. 
By  the  conditions, 
X  +  2^  +  z  =  8, 
a:  +  1/  —  4  =  2, 

100  a:  +  10  y  +  2  +  99  =  100  z  +  10  2/  +  a:. 
Solving  these  equations,  z  —  %  y  =  5,  x  —  1. 
Therefore,  the  number  is  152. 

Note  1.  In  verifying,  the  results  should  be  tested  directly  by  the  conditions 
of  the  problem.  Thus,  in  the  above,  the  sum  of  2,  5,  and  1  is,  as  one  condition 
requires,  8.  The  sum  of  1  and  5  exceeds  2  by  4»  The  sum  of  162  and  99  is  251 
fiB  required. 


PROBLEMS.  241 

15.  A  number  is  expressed  by  three  digits.  The  middle 
digit  is  twice  the  left  hand  digit,  and  one  less  than  the 
right  hand  digit.  If  297  be  added  to  the  number,  the 
order  of  tlie  digits  will  be  reversed.     Find  the  number. 

16.  A  number  is  expressed  by  three  digits.  The  sum 
of  the  digits  is  18 ;  the  number  is  equal  to  99  times  the 
sum  of  the  first  and  third  digits,  and  if  693  be  subtracted 
from  the  number,  the  digit  in  the  units'  and  hundreds' 
place  will  be  interchanged.     Find  the  number. 

17.  The  sum  of  the  three  digits  of  a  number  is  n ;  the 
number  is  equal  to  a  times  the  sum  of  the  first  and  third 
digits,  and  if  m  be  subtracted  from  the  number,  the  digit 
in  the  \mits'  and  hundreds'  place  will  be  interchanged. 
Find  the  number. 

18.  If  a  certain  number  be  divided  by  the  sum  of  its 
two  digits  the  quotient  is  3,  and  the  remainder  3 ;  if  the 
digits  be  interchanged,  and  the  resulting  number  be  di- 
vided by  the  sum  of  the  digits,  the  quotient  is  7,  and  the 
remainder  9.     Find  the  number. 

19.  If  a  certain  number  be  divided  by  the  sum  of  its 
two  digits  the  quotient  is  a,  and  the  remainder  b ;  if  the 
digits  be  interchanged,  and  the  resulting  number  be  di- 
vided by  the  sum  of  the  digits,  the  quotient  is  c,  and  the 
remainder  m.     Find  the  number. 

20.  The  sum  of  the  three  digits  of  a  number  is  16.  If 
the  number  be  divided  by  the  sum  of  its  hundreds'  and 
units'  digits  the  quotient  is  77  and  the  remainder  6 ;  and 
if  it  be  divided  hy  the  number  expressed  by  its  two  right- 
hand  digits,  the  quotient  is  16  and  the  remainder  5.  Find 
the  number. 


242  ELEMENTS  OF  ALGEBRA. 

21.  The  sum  of  the  three  digits  of  a  number  is  9.  If 
the  number  be  divided  by  the  difference  of  its  hundreds' 
and  units'  digits,  the  quotient  is  157,  and  the  remainder  1; 
and  if  it  be  divided  by  the  number  expressed  by  its  two 
right-hand  digits,  the  quotient  is  21.     Find  the  number. 

22.  A,  B,  and  C  can  together  do  a  piece  of  work  in  12 
days ;  A  and  B  can  together  do  it  in  20  days ;  B  and  C 
can  together  do  it  in  15  days.  Find  the  time  in  which 
each  can  do  the  work. 

Suggestion.     Let  x  =  the  number  of  days  in.  which  A  can  do  it, 

and  y  =  the  number  of  days  in  which  B  can  do  it, 

also  2  =  the  number  of  days  in  which  C  can  do  it. 

1111111  111 

Theequationsare-  +  ^  +  ~  =  12'   ^  +  2^=  20'  ^^^  y  +  z  =  \b'^ 

from  which  a;  =  60  and  y  =  2  =  30. 

23.  A  and  B  can  do  a  piece  of  work  together  in  48 
days ;  A  and  C  in  30  days ;  B  and  C  in  26|  days.  How 
many  days  will  it  take  each,  and  how  many  altogether,  to 
doit? 

24.  A  and  B  can  do  a  piece  of  work  together  in  a  days; 
but  if  A  had  worked  m  times  as  fast,  and  B  n  times  as 
fast,  they  would  have  finished  it  in  c  days.  How  many 
days  will  it  take  each  to  do  it  ? 

25.  A  drawer  will  hold  24  arithmetics  and  20  algebras; 
6  arithmetics  and  14  algebras  will  fill  half  of  it.  How 
many  of  each  will  it  hold  ? 

26.  A  purse  holds  19  crowns  and  6  guineas ;  4  crowns 
and  5  guineas  fill  JJ  of  it.  How  many  will  it  hold  of 
each  ? 


PROBLEMS.  243 

27.  A  purse  holds  c  crowns  and  a  guineas ;  ci  crowns 

tn 
and  ai  guineas  will  fill  —  th  of  it.     How  many  will  it  bold 

of  each  ? 

28.  A  and  B  together  could  have  completed  a  piece  of 
work  in  15  days,  but  after  laboring  together  6  days,  A  was 
left  to  finish  it  alone,  which  he  did  in  30  days.  In  how 
many  days  could  each  have  performed  the  work  alone  ? 

29.  Two  persons,  A  and  B,  could  finish  a  piece  of  work 
in  m  days;  they  worked  together  a  days  when  B  was 
called  off  and  A  finished  it  in  n  days.  In  how  many  days 
could  each  do  it  ? 

30.  A  can  row  8  miles  in  40  minutes  down  stream,  and 
14  miles  in  1  hour  and  45  minutes  against  the  stream. 
Find  the  number  of  miles  per  hour  that  the  stream  flows, 
also  that  A  rows  in  still  water. 

Suggestion.     Let  x  =  the  number  of  miles  per  hour  that  A  can 
row  in  still  water, 
and  y  =  the  number  of  miles  per  hour  that  the 

stream  flows. 
Then,  x  +  y  =  the  number  of  miles  per  hour  that  A  can 

row  down  the  stream, 
and  X  —  y  =  the  number  of  miles  per  hour  that  he  can 

row  up  the  stream. 
Since  the  distance  divided  by  the  rate  will  give  the  time,  by  the 
conditions, 

8 2 

x  +  y~  3* 

31.  A  can  row  m  miles  in  h  hours  down  stream,  and  mi 
miles  in  ^i  hours  against  the  stream.  Find  the  number  of 
miles  per  hour  that  the  stream  flows,  also  that  A  rows  in 
still  water.     Apply  the  result  to  problem  30. 


244  ELEMENTS  OF  ALGEBRA. 

32.  A  boatman  sculls  down  a  stream,  which  runs  at  the 
rate  of  5  miles  an  hour,  for  a  certain  distance  in  3  hours, 
and  finds  that  it  takes  him  13  hours  to  return.  Find  the 
distance  sculled  down  stream,  and  his  rate  of  rowing  in 
still  water. 

33.  A  man  who  can  row  at  the  rate  of  15  miles  an  hour 
down  stream,  finds  that  it  takes  3  times  as  long  to  come 
up  the  stream  as  to  go  down.  Find  the  number  of  miles 
per  hour  that  the  stream  flows. 

34.  A  waterman  rows  30  miles  and  back  in  12  hours ; 
and  he  finds  that  he  can  row  3  miles  against  the  stream 
in  the  same  time  as  5  miles  with  it.  Fiud  the  number  of 
hours  in  going  and  coming  respectively ;  also,  the  number 
of  miles  per  hour  of  the  stream. 

35.  A  waterman  can  row  down  stream  a  distance  of  m 
miles  and  back  again  in  h  hours ;  and  he  finds  that  he  can 
row  h  miles  against  the  stream  in  the  same  time  he  rows 
a  miles  with  it.  Find  the  number  of  hours  in  going  and 
coming,  respectively ;  also  the  number  of  miles  per  hour 
of  the  stream,  and  his  rate  of  rowing  in  still  water. 


36.  Five  pounds  of  sugar  and  3  pounds  of  tea  cost 
$2.05,  but  if  the  price  of  sugar  was  to  rise  40  %,  and  the 
price  of  tea  20  %  they  would  cost  $2.51.  Find  the  num- 
ber of  cents  in  the  cost  of  a  pound  of  each. 

37.  If  /  pounds  of  sugar  and  h  pounds  of  tea  cost  m 
dollars,  and  the  price  of  sugar  was  to  rise  a  % ,  and  the 
price  of  tea  h  %,  they  would  cost  n  dollars.  Find  the  num- 
ber of  cents  in  the  cost  of  a  pound  of  each. 


PROBLEMS.  245 

38.  The  amount  of  a  sum  of  money,  at  simple  interest, 
for  11  months  is  S1055;  and  for  17  months  it  is  S1085. 
Find  the  sum  and  the  rate  per  cent  of  interest. 

39.  The  amount  of  a  sum  of  money,  at  simple  interest, 
for  VI  months  is  a  dollars  ;  and  for  n  months  it  is  h  dollars. 
Find  the  sum  and  the  rate  of  interest. 

40.  A  grocer  mixes  three  kinds  of  coffee.  He  can  sell 
a  mixture  containing  2  pounds  of  the  first  kind,  9  pounds 
of  the  second,  and  5  pounds  of  the  third,  at  18  cents  per 
pound  ;  or  one  composed  of  6  pounds  of  the  first,  6  pounds 
of  the  second,  and  9  pounds  of  the  third,  at  19  cents  per 
pound ;  or  one  composed  of  5  pounds  of  tlie  firet  kind,  2 
pounds  of  the  second,  and  18  pounds  of  the  third,  at  22 
cents  per  pound.  Find  the  number  of  cents  in  the  cost  of 
a  pound  of  each  kind. 

41.  The  fore-wheel  of  a  carriage  makes  6  revolutions 
more  than  the  hind-wheel  in  going  120  yards ;  if  the  cir- 
cumference of  the  foie  wheel  be  increased  by  J  of  its  pres- 
ent size,  and  the  circumference  of  the  hind-wheel  by  J  of 
its  present  size,  the  0  will  be  changed  to  4.  Find  the 
number  of  yards  in  the  circumference  of  each  wheel. 

42.  The  fore-wheel  of  a  carriage  makes  a  revolutions 
more  than  the  hiud-wheel  in  going  b  feet.     If  the  circum- 

ference  of  the  fore-wheel  be  inci-eased  by  —  th  of  itself,  and 

8  ^ 

that  of  the  hind-wheel  by   -  th  of  itself,  the  hind-wheel 

r 

will  make  c  revolutions  more  than  the  fore-wheel.     Find 
the  circumference  of  each  wheeL 


246  ELEMENTS  OF  ALGEBRA. 

43.  A  grocer  has  two  kinds  of  coffee.  He  sells  a  pounds 
of  the  first  kind,  and  h  pounds  of  the  second,  for  m  dollars; 
or,  ax  pounds  of  the  first  kind,  and  hi  pounds  of  the  second, 
for  mi  dollars.  Find  the  number  of  dollars  in  the  price  of 
a  pound  of  each  kind. 

44.  A  jeweller  has  two  silver  cups,  and  for  the  two  a 
single  cover  worth  90  cents.  If  he  puts  the  cover  upon 
the  first  cup  it  will  be  worth  1^  times  as  much  as  the 
other ;  if  he  puts  it  upon  the  second  cup  it  will  be  worth 
lyig  times  as  much  as  the  first.  How  many  dollars  in  the 
value  of  each  cup  ? 

45.  A  jeweller  has  two  silver  cups,  and  for  the  two  a 
single  cover  worth  a  dollars.  If  he  puts  the  cover  upon 
the  first  cup,  it  will  be  worth  m  times  as  much  as  the 
other ;  if  he  puts  it  upon  the  second  cup  it  will  be  worth 
n  times  as  much  as  the  first.  How  many  dollars  in  the 
value  of  each  cup  ? 

46.  A  broker  invests  $5000  in  3's,  $4000  in  4's,  and 
has  an  income  from  both  investments  of  $315.50.  If  his 
investment  had  been  $1000  more  in  the  3's,  and  less  in 
the  4's,  his  income  would  have  been  $5.50  greater.  Find 
the  market  value  of  each  class  of  bonds. 

Note  2.  3's  means  bonds  which  bear  3  %  interest.  The  "  quoted  "  price  of 
a  bond  is  its  market  value.  Thus,  a  bond  quoted  at  115i  means  that  a  $100 
bond  can  be  bought  for  $115.50  in  the  market. 

47.  A  broker  invests  m  dollars  in  a's,  n  dollars  in  c's, 
and  has  an  income  from  both  investments  of  h  dollars.  If 
his  investment  had  been  d  dollars  less  in  the  a's,  and  more 
in  the  c's,  his  income  would  have  been  p  dollars  less.  Find 
the  price  paid  for  each  kind  of  bonds. 


PROBLEMS.  247 

« 

48.  A  and  B  do  a  piece  of  work  together  in  30  days, 
for  which  they  are  to  receive  $1G0.  But  A  is  idle  8  days 
and  B  is  idle  4  days,  in  consequence  of  which  the  work 
occupies  5J  days  more  than  it  would  otherwise  have  done. 
Find  the  number  of  dollars  received  by  each. 

49.  A  and  B  do  a  piece  of  work  together  in  m  days,  for 
which  they  are  to  receive  c  dollars.  But  A  is  idle  a  days 
and  B  is  idle  h  days,  in  consequence  of  which  the  work 
occupies  n  days  more  than  it  would  otherwise  have  done. 
Find  the  number  of  dollars  received  by  each. 

50.  The  amount  of  a  sum  of  money,  at  simple  interest, 
for  5  years  is  S600;  and  for  8  years  it  is  $660.  Find  the 
number  of  dollars  in  the  sum,  and  the  rate  of  interest. 

51.  The  amount  of  a  sum  of  money,  at  simple  interest, 
for  a  years  is  m  dollars ;  and  for  h  years  it  is  n  dollars. 
Find  the  number  of  dollars  in  the  sum,  and  the  rate  of 
interest. 

52.  If  a  grocer  sells  a  box  of  tea  at  30  cts.  a  pound,  he 
will  make  SI,  but  if  he  sells  it  at  22  cts.  a  pound,  he  will 
lose  S3.  Find  the  number  of  pounds  in  the  box,  and  the 
number  of  cents  in  the  cost  of  a  pound. 

53.  The  smaller  of  two  numbers  divided  by  the  larger 
is  .21,  with  a  remainder  .04162.  The  greater  divided  by 
the  smaller  is  4,  with  .742  for  a  remainder.  Find  the 
numbers. 

54.  The  smaller  of  two  numbers  divided  by  the  larger 
is  a,  with  a  remainder  m.  The  greater  divided  by  the 
smaller  is  h,  with  c  ibr  a  remainder.     Find  the  numbers. 


248  ELEMENTS  OF  ALGEBRA. 

CHAPTEK   XVIII. 

EXPONENTS. 

97.    An  Exponent  is  a  figure  or  term  written  at  the  right 
of  and  above  a  number  or  term  (Art.  21). 

'^  m 

Thus,  in  the  expressions  5^,  a%  6",  and  (a  +  by;  2,  c,  —,  and  3 

are  exponents. 

Zero  Exponents.     When   the    dividend  and  divisor  are 
equal  the  quotient  is  1. 

Thus,  ^,=  1;  -,=  1;    ~,=  i;    ~  =  I;    etc. 

But  (Art.  30),  32  =  32-2  =  30;  ^  =  aO;  ^^  =  «^  ^  =  «^  etc. 
Therefore,  it  follows  that  a^  =  I.     Hence,  in  general, 
I.   Any  expression  with  zero  for  an  exponent  is  1. 

The  Reciprocal  of  a  number  is  unity  divided   by  that 
number. 

Thus,  the  reciprocal  of  n  is  -;  of  n  +  m  is 


n  -\-  m 

Negative  Integral  Exponents. 


a3  X  a-8=a8-3  =  «o^i 

Divide  by  a^, 

a»  X  a-«  =  «"-«  =  a^=  \ 

Divide  by  a", 

a~**  =  — .     Hence,  in 

EXPONENTS.  249 

II.    A  negative  integral  exponent  indicates  the  reciprocal 
of  the  expression  with  a  corrcspoiuling  positive  exponent. 

The  expression  a",  where  n  is  any  positive  integer,  represents  tht 
product  o/n  equal  factors,  each  equal  to  a.     It  has  been  shown  that : 

Art.  21,  a^  X  a*  —  a*»+". 

Art.  30,  a"^  —  a^  =  a"*-",  where  m  is  greater  than  n. 

Art.  30,  ti^  -T  a'*=    ^_,^,  where  m  is  less  than  n. 

Art.  27,  (a*")"  =  a*"",  whatever  the  value  of  m. 

Thus, 

By  Art.  21,  a*  X  a"  X  a'  x  .. . .  a"*  =  a»+«+ ''+••'». 

Take  n  factors  of  a*,  a*,  a**, a"*,  and  suppose  each  of  the  n  ex- 
ponents equal  to  m,  then  it  follows  that 

(amy  _  Qtnn^     Hencc,  m  can 
be  positive  or  negative,  i  itegral  or  fractional. 

By  II..  <'-"  =  ii- 

a"* 
Multiply  by  a"*,  a-"  X  a"»  =  —  • 

If  m  is  greater  than  n,  Art.  30,  -;;  =  a*"-". 
Therefore,  a-"  >^  a*"  =  a"*-*. 


If  m  is  less  than  n, 


a*         1 


a»      a 


i«  — w 


By  II.,  7=^  =  «' 


a" 


Therefore,  a-»  x  a*"  =  «"•-'•  for  all  possible  integral 

values  of  m  and  n. 


98.    By  Art.  27,  (ai)"  =  a"  X  6". 

Therefore,  o-^*"  =  (aft)*. 

Similarly,  a"  X  ft"  X  c"  X  . . . .  p"  =  (a  6  c  . . . .  />)■ 


250  ELEMENTS   OF  ALGEBRA. 

If  n  is  a  negative  integer, 


«-"  X  *-"  =  a«  X  6"  =(«6)»  =("*)-• 

Similarly, 

a-«  X  i-^  X  c-»  X  ....jD-"=  {ahc  ... 
general. 

.  Py- 

Hence,  in 

I.  The  product  of  tv:o  or  more  factors,  each  affected  with 
the  same  exponent,  is  the  same  as  their  product  affected  wiih 
the  exponent. 

By  IL,  Art.  97,  a"  -f  &«  =  a«  6-». 

Also,  a«  6-«  =  (a  6- 1)«  =  (? ) • 

Therefore,  a**  -f-  ft"  =  [  r 

Similarly,  a-" -^  &-»»  =  f  r)  Hence,  in  general, 

II.  TAe  quotient  of  any  two  factors,  each  affected  with 
the  same  exponent,  is  the  same  as  their  quotient  affected  with 
the  exponent. 

Illustrations:  I.  22  X  3^  =  (2  X  3)2  =  (6)2  =  36  ;  28  X  3^  X  48 
=  (2  X  3  X  4)8  =  (24)8  =  13824  ;  2-2«  X  3-2«  X  4-2"  =  (2  X  3  X  4)-2  « 
=  [(24)2]-"  ^  (576)-«=  ^ ;  (f)-2  X  (f)-=^X  (i)-2z.  (|  x  f  X  i)-2 


1 


(})-  =  ^=16. 


16\-8 


II.    242-^62=  (5^4)2=  (4)2:^16;  (_16)-8--(-4)-8=(--^) 


1\4ot  /'1\4'»  1 


These  examples  are  said  to  be  simplified,  that  is,  they  are  expressed 
in  their  simplest  forms. 


EXPONENTS.  261 

Exercise  90. 
Simplify : 

1.  (n^f  X  {a^f  X  (71)2;     (J)2  X  (2)2  X  (§)2  -  (if. 

2.  (a;*  y-'")8 -f- (oj- V")^    (216  2^2)4^(54  2^-2)4^ 

3.  (|a)8  X  (|aj-2)8;    (^ri)-"  X  (^■)-;    (a;)*  x  (2:"^)*. 

4.  (x)"  X  {a^Y;  (f)-"  X  (})-"  X  (2)"";  {a-Hf  X  (a6-8)6 

5.  (2  71)10  X  (2- im)iO;    (a  6-ic-2)3  -  (a-ife-ac-*?/^)^. 

6.  (4a*^a:«)-"^(2-2a-3*2;-V)"'*;   (^-iy*)-^-^(^^rV. 

7.  a---  X  (3  2>"')-"  X  (ci)-™;    (:r)i-  -  (^'y-". 

8.  («-2  5)-2  X   («  ?>-3)-2;     (rtS  J8  +  ^6)- 3  ^  (a6-an3)-8 

9.  (i)"  X  {^T  X  (J)"  ;    (a2''  +  a"  ?»2'')-i  x  («"  -  &2«)-i. 

10.  («-i)-^  X  (xi)-5  X  (x-t)-6  X  (at) -5  X  (&i)-^ 

11.  (^^"+*)''  X  (i2-*j"  X  (an)"  X  {b-nY;  aS  -^  (2  a)8 

12.  «  2x(2a)-2-f.^^y;  (2«-2)-2x^^y'x(|a)-2. 

13.  (a-i  V^x)-'  X  (.r-2v^6)-8.    (,,i)2''  X  (2^)2"  X  (c")2". 

14.  (2")-"  X  (2"-!)-''  X  (2-2—1)—  X  (2-2"+i>-'»  X  (r)-\ 

15.  (2''+i)"'  X  (2— "+")"•  X  (2"'-!)'"  X  (4-"-^)'"  -T-(ir))-'". 

16.  [(2:-y)-8]-X[(rr  +  7/)-]-8;    (|)-"  x  ff)- -  (J)- 


252  ELEMENTS  OP  ALGEBRA. 

99.  Positive  Fractional  Exponents.  If  m  and  n  are  both 
positive  integers, 

Kan)   =  a"*. 

m  

Take  the  nth  root  of  both  members,  a'*  =  \/a^. 

m 

Therefore,  a«  means  the  nth  root  of  the  mth  power  of  a,  or  the 
mth.  power  of  the  nth  root  of  a.     Hence, 

The  numerator,  in  a  fractional  exponent,  denotes  a  power, 
and  the  denominator  a  root. 

The  denominator  of  the  exponent  corresponds  to  the  index  of  the 
root.     Thus,  (81)1  =  \/{Siy  =  (^8iy  =  (3)3  =  27. 

m  

In  a«  =  /y/a"*,  m  is  the  index  of  the  power,  and  n  is  the  index  of 
the  root ;  also  a,  m,  and  n  may  be  any  numbers.  The  expression 
may  be  raised  to  the  power  indicated  by  the  numerator  of  the  expo- 
nent and  then  extract  the  root  of  the  result  indicated  by  the  denomi- 
nator; or,  extract  the  root  first  and  then  raise  the  result  to  the  power 
indicated  by  the  numerator  of  the  exponent.     Thus, 

(-8)1  -  -V^FS?  =  v'64  =  4  ;  or,  (-  8)f  =  (-v^^)'  =  {~  ^Y  =  4- 

Notes:    1.  a-«  is  read  "a  exponent  —n;"  a"  is  read  "a  exponent  -; 
a~  «  is  read  "a  exponent ."    These  are  abbreviated  forms  for  "a  with  an 

exponent  —n;      etc. 

m 

2.  It  is  manifestly  incorrect  to  read  a«  "  the  -  th  power  of  a."  There  is 
no  such  thing  as  a  fractional  power. 

3.  We  must  be  careful  to  notice  the  difference  between  the  signification  of  a 
fraction  used  as  an  exponent,  and  its  common  signification.  Thus,  f  used  as 
an  exponent  signifies  that  a  number  is  resolved  into  five  equal  factors,  and  tlie 
product  of  four  of  them  taken. 


mXc         mc 


100.    By  Art.  73,  a»»  =  a«  x  "  =  anc; 

m 

e 
m  7H-T-C  n 

also,  a**  =  a"  ^  "  =  a^     Hence, 


EXPONENTS.  263 

L   Multiplying  or  dividing  the  terms  of  a  fractional  ex- 
ponent hy  the  same  number  will  not  change  the  value  of  the 


expression. 

^n^a^-"^. 

But 

a^^^=^a^, 

and 

^a^=^/^a. 

Therefore, 

Jrn  -  \/;^a. 

Hence,  in  general, 

II.  The  mnth  root  of  a  number  is  equal  to  the  mth  root 
of  the  nth  root  of  that  number. 

niuBtrations. 

2*  =  2*  ;  6»  =  6i  ;  62«  =  6«^;  ^^64  =  '^  ^64  =  ^  =  2. 

101.  Negative  Fractional  Exponents.  If  m  and  ti  are 
both  positive  integers, 

(  --Y 

\a   »•/   =  a""*. 
By  II.,  Art  97,  a-«  =  ^. 

Take  the  nth  root  of  both  members, 

m  1 

a   »  =  — .    Hence, 

Ajiy  expression  affected  with  a  negative  fractional  expo- 
nent is  equal  to  the  rccipror/U  of  the  expression  with  a  cor- 
responding positive  exponent. 

_"*      1       "•       1 
Notes :  1 .  From  the  relation  a    »•  =  -^ ,  a"  =  — ^ .    Hence,  the  method  of 

o*  a    * 

Art.  30  is  true  for  fractional  exponents. 

2.  Any  factor  of  the  dividend  may  be  removed  to  the  divisor  (or  from  the 
numerator  to  the  denominator  of  a  fraction),  or  any  factor  of  the  divisor  to 
the  dividend,  hy  changing  the  sign  of  its  exi>onent. 


254  ELEMENTS  OF  ALGEBRA. 

Illustrations.  2^^  =  ^  =  ^^1  (1)"^  =  (^  =  |  =  I'  ^  =  «"^ 

,,.-1         -3  1  1  1         2*  X   3        3       /  u^ 

X  (i)    '  X  4    '  =  ^,  X   I   X  -3  =  -j,~  =  -     t-i  ^x-")~ 

1        11:^, 
-^       -^      -  x^  '  x~  x^-^       • 

102.  (ah^y=ab. 

Take  the  nth  root  of  both  members, 

J        1       1  1  j^ 

Similarly,*     an  x  &"  X  c«  X  . . . .  ;?"  =  (a  6  c  . . . .  ;?>.     Hence, 

I%e  product  of  two  or  more  factor's  each  affected  with  the 
sa7ne  root  index,  is  the  same  as  their  product  affected  with  the 
root  index. 

In  the  same  manner  we  can  prove  that 

Kotes :  *  1.  If  we  suppose  that  there  are  m  factors  oi  a,h,  c, p,  and  that 

each  factor  is  equal  to  a,  then  it  follows  that 

By  Art.  99,  [a^Jn  =  an. 

Therefore,  Va»/    =  an. 

2.   Similarly,  \an)    =  a.        Hence, 

Tfie  nth  power  of  the  nth  root  of  a  number  is  equal  to  that  number. 

Illustrations.     (A)*  X  (f,)^  X  8*  =  (f  X  |       8)*  =   ^'^  =  |; 


EXPONENTS.  255 


»v 


103.  (a"  X  a'T'  =  (j««+»*. 

m  b  JL 

Take  the  ncth  root,  a«  X  a«  =  (0™'+"*) 


m  .  6 


By  Art.  99,  (a"» «+"«>)"''  =  a" ''^ «. 

m  6  m      6 

Therefore,  a»Xae  =  a«'^«.    • 

iw  ft  r  t  ?j.*j.'"i. ? 

Similarly,    a*  X  a«  X  a»  X    •  •    a«  =  a»    «    i"*"""  «.     Hence, 

I.  Th£  product  of  several  expressions  consisting  of  the 
same  factory  affected  with  any  exponent,  is  the  factor  with 
an  exponent  equal  to  the  sum  of  the  exponents  of  the  factors. 

By  Arts.  101,  21,  c* -f- ac  =  a*  X  a    «  =  a»»    «.     Hence, 

II.  The  quotient  of  two  expressions  coTisistiiig  of  the  same 
factory  affected  with  any  exponent,  is  the  factor  with  an 
exponent  equal  to  that  of  tlic  divideiid  mimis  tJiat  of  the 


lUuBtrationa :     I.     5*  X  5"^  X   5  =  5*"*"^*  =  5*  =  >^125 ; 
X*  X  a:*  X  a:"  =  x"+*. 

II.   2*-r2*  =  2^"*  =  2i  =  v^2;  (a  +  6)' -^  (a+ 6)*  =  («  +  6)'"* 
=  (0  +  6)- A. 

Exercise  91. 

Simplify : 

1.  16-f  X  16-i;    25-i  X  25i;    3^  x  (^;  a"!  X  ^. 

n  n  6_»»  X  2  m 

2.  aixa^Xa^;    n-^' X  n   ';    m    "Xw    '';  2-iV2. 

3.  y"^  X  7/"  »  ;    a'  -T-  a~';    rt*  X  J;   (a^)*  -r-  (a^)h 

4.  (-2)-i-(-32H;    a* -at;    (^-J  -  (..y)-i. 


256  ELEMENTS  OF   ALGEBRA. 

5.  x^  ^  rr2«;    a    '^-^m     ^  ;    (a-  h)^  -  (ah  +  hi)i. 

6.  32«  -^  3" ;    (a  -  &)"  ^(a-b);    (x-^/2)h  ^  (//  2;^)?. 

\a^b  ^J         \a-^b^J        \ax-^  J         \x   ^ J 
8.    {a -2  xf  x{a-2xf  x{2x-afx{ci-2  x^. 

10.    (.r  +  yy-''  ^{x  +  ^)-";    a3^+2y  _^  a2^-3^;    lA-^rA. 

/        m     \      ^  /  »«  — 1        \  —1—  /     \  "'  +  ^"  /     \  ^ 

11        1*^         Xmnp  /  3?  \mn/)  f  CC\       n  f  CC\m 

13.   at  -=-  ai;    2"  x  (2»)»-'  x  2"  +  '  X  2»-i  x  4-". 

^^2  .  i^y     {off     (x'f  .    ,  ;   ..  ,  ^  /^V''" 

X04.        caa"'=(a")-=a-. 

Take  the  n  qih.  root  of  the  first  and  last  members, 

(m    r    (  \p  mp    rp    tp 

The  principles  of  this  chapter  are  true,  whatever  the  values  of 

a,h,c,....m,  n,  p,  and  q  ;  that  is,  a,b,c, m,  n,  p,  and  q  can  be 

positive  or  negative,  integral  or  fractional. 


EXPONENTS.  257 

niustrations.     (2'  X  3*  X  4"^)'  =  s'""*  X  S*""'  X  4-*'**  =  2« 

X  2*  X  3*  X  4-1  =  ^2  X  V3  ;    •  L(a"^)"«]fl^ -r-  I  [(aW^JJlJ 


N      m 


a"«    »•  =  a-*  —  a  =  a. 

105.   Negative  and  Fractional  Root  Indices. 

_  j>i_       _w       \  1 

an       V« 

m  _m 

_  e  _£  £  1  1 

Similarly,      y^a™  =  a»«  =  a    «  =  —  =  "^TZ*     Hence, 

^  negative  root  index,  either  integral  or  fractional^  indi- 
cates iJie  reciprocal  of  Uie  expression  with  a  corresponding 
positive  index. 

Note.  Since  it  is  impossible  to  extract  a  fractional  or  negative  root,  or  raise 
an  expression  to  a  fractional  or  negative  power,  in  order  to  perform  the  opera- 
tion indicated  by  such  indices  some  preliminary  transformations  must  be  made. 

lUuBtrationB.  ~i/^  =  -: —  =  -|  =  -r  =  —^  ;     i/4a«  =  (4  ay 

_  I  _  ±    ^_  _  ±  _  J-  =    1 

a* 

Exercise  92. 
Simplify : 

1.  1^27;     V^;    |/32  m-iO;    Vsla-^;    V^. 

2.  1^8;    [(63)2(a*)8(6-8)(a-6J-i)2]6;    ^8a*6— ic"-2 

17 


258  ELEMENTS   OF  ALGEBRA. 

2 

V  ^  /  \y'J  Kni^n^J  '     vo«'"^'*^"^'    V25' 

1    1^      /     J_  \a2  -  62 

Queries.  What  does  a  negative  exponent  indicate  1  A  fractional 
exponent  ?  A  negative  fractional  root  index  ?  Any  expression  with 
0  for  its  exponent  =  ?  Why  ?  What  is  the  product  of  as  and  a^  1 
Prove  it. 

Miscellaneous  Exercise  93. 

Express  with  fractional  exponents  and  negative  power 
indices : 

1.    ~\^^;    ~^;     4"^;    'V^;    (^a)^;     ^^5^2 

m 

Express  with  radical  signs  and  negative  integral  root 
indices  : 

m  O 

3.    a-t;     a^hic~^;    4:ah~^;    7a~^x~'';    — -  . 

X    4 


EXPONENTS.  259 

Express  with  radical  signs  and  fractional  root  indices : 


n     n 


9  1  m     m      X  T7"« 

4.   at;    (4a2)!;    alz^U;    a'"5";    a"  ft"; 


xy. 


Express  with  fmctional  exponents  and  fractional  power 
indices : 

5.    ^Jbi;    y/2'6-;    3v'(8a-8)-|;  ^a'~^;    Va;    ^^5. 

Express  in  the  form  of  integral  expressions : 

Sa^b       5         m*  a  x~^        x~^  a^ 


c-2  '    ahc'    n-r    4^1'    4^^'    -^-|'    aib-l 

if 

Express  with  literal  factors  transposed  from  the  numera- 
tors to  the  denominators: 

Simplify  and  express  with  positive  exponents : 

8.  4^;   v^^^a.  yj(h^;  "-^/^;  «-»;  [Va^  H- V^]"". 

_    2  ai  X  3  a-i     a  2^  x  a'^  x  v"^      «/ ^        s.-y 

9.  -== ;    —=r. — ^^  ;    Vm-3 -^  V7/il 

10.  2;-ix2a;-i;    (^V*;    a^  x  «i  X  a-J. 

11.  -^4^;    (j^y^;    V^^h  X  ^J'^F-s 

12.  aUiaxa-i6-U-i;     (f^)"';    y/^- 


260  ELEMENTS  OP  ALGEBRA. 

13.  aH^c^  Xa-H-^c-i;    \l^\    i/(^~H*)^. 

14.  y"  X  y  X   ";  (x  +  y)^  X  n^  X  n~r^  X  Vti. 

15.  Y/(m^'^V';  y/«»6"i  i(|)"';  -tF^. 

16.  (77i-i\^a)-3  X  'v/(a-2  V^";  y^^'  +  — 

17.  VaF^W^'-^{ah~~^y;    l^^ll^. 


2"'  2»2 


0  a;  '■m  "^  n 


^^-  m2  ^   m-w   Ml    ,  J'    (m2-7^2)2 
2"(2"-i)"    1    2"  +  ^     4""^^ 


2»+i  X  2"-"   4^"'  (22»)"-i  •  (2"-i)"  +  > 

(9"x32x5kV27» 

V /  / ;  (2"  X  S"*)""  Is"  X  6*"")"' 


23.  pir^^ 


EXPONENTS.  261 

Multiply : 
24   a*fe~i  -  a^h"^*  +  1  by  a^h~^'  +  1. 

25.  a^  +  a2»&*'  +  h^'  by  a"  —  a^"^)*"*  +  ^'. 

26.  a^h    '—a* 5   •+«   *7>i— a    »6«  by  a«6  i  +  a   "ft*. 
Divide : 

27.  a^  +  rt^  M"'  +  &•  by  a»  +  a2^  M"'  +  ft*"., 

28.     a,-*"*"-*)  —  y2m(m-l)    ^y    ^{n-l)^^(m-« 

29.  a;**"*  -  if^-*  by  ar"*"''  +  i/^'"-^ 

30.  a;^"'-*"  -  ^m'-am  by  ^"^-^"^  ±  f*-'*, 

31.  a^ —3^+4:a*'*x*'—4:a^x^'  by  a2*  + 2a^2:<''  — .r^*. 

32.  a3+a~i*  by  a5  +  rt-^;  riT/i  +  mx^  by  n^yi  +  w^a;^. 
Separate  into  two  factors  : 

33.  a-^-b;    a'^  -  ft-f ;    aV  -  6-2«. 

Expand : 

34.  (a-U-6-ia:)*;    (a:-2a;-i)8;     [(a"^ -«?)']". 

Resolve  into  prime  factors,  and  find  the  products  of: 

35.  N?'!^,  4^2,  \/96,  \^ 

36.  ^12,  v^72,  \^,  ^,  ^^^576,  V2l 


262  ELEMENTS  OF  ALGEBRA. 

Find  the  cube  roots  of  : 

38.  8  a-2  -  12  a- V-  +  6  a-f  -  a"! 

39.  a;3__9^+27a:-i-27ar-3. 
Find  the  6th  roots  of : 

40.  a;«  +  ^  -  6  (a;*  +  ^)  +  15  (^x^  +  i)  -  20. 

41.  729  -  2916  a2«  +  4860  «*"  -  4320  a^"  +  2160  a^" 

-576^10"  +  64a^2«, 

42.  a;- 12  -  6  :r- 10  +  1 5  a;-  8  -  20  ^^-  6  +  1 5  ic-  4  -  6  aj- 2  +  1. 
Simplify  and  express  with  positive  exponent  : 


!i+i,  ifan  i/4  X  4"-i 

H:i, 

y/4«-i    >^  4„  +  i 


4fi  ^"^"^  [(8a-6?>)2"]5"       Q 

9(a;0  +  2/0  +  ^)-2m3'    [(4a-3  6)5"]2'''    ^  +  T • 

.,,    (20a3H8a;2?/2-12y^)"     (m^  +  ^^)^  (^^  -  n^)^ 
[4  (2:2  +  2/2).f    '      '  m6--?i6 


RADICAL  EXPRESSIONS.  263 

CHAPTEE  XIX. 
RADICAL  EXPRESSIONS. 

106.  A  Surd  is  an  indicated  root  that  cannot  be  exactly- 
obtained  ;  as,  V5 ;  'V^f ;  \^a^. 

The  Order  of  a  surd  is  indicated  by  the  root  index. 

Surds  are  said  to  be  of  the  second^  third y  fourth,  etc.,  or  nth  order, 
according  as  the  second,  third,  fourth,  etc.,  or  nth  roots  are  retjuired. 
Thus,  'v/a,  ^a,  \/b,  etc.,  yx,  are  quadratic,  cubic,  biquadratic, 
etc. 

Surds  are  of  the  same  order  when  they  have  the  same  root  index ; 
as,  ^b,  ^a\  and  ^¥. 

A  surd  is  in  its  simplest  form  when  the  expression  un- 
der the  radical  sign  is  integral,  and  in  the  lowest  degree 
possible  ;  as,  ^32  a*  =  \/2^  a^  x  4  a  =  2  a  v^4  a. 

Similar  or  Like  Surds  are  those  which,  when  reduced  to 
their  simplest  forms,  have  the  mme  surd  factor  ;  as,  3  \/3 
and  a/3  ;  2  a  vh  and  c  ^/h.  Otherwise  the  surds  are 
dissimilar. 

Hotes :  1 .  When  a  surd  is  expressed  by  means  of  the  radical  sign,  it  is 
called  a  Badical  ExpressioxL 

2.  An  Irrational  Expression  is  one  which  involves  a  surd  ;  as,  V3 ; 
a  -\-h  \c^. 

3.  An  indicated  root  may  have  the  form  of  a  surd,  without  really  being  a 
8uixl.     Tims,   Vi  and   Va»  have  the  f(rrm  of  surds. 

4.  Rational  factors  or  expressions  are  those  which  are  not  surds  ;  as,  2; 
a*x  —  bf^y. 

5.  Since  a"  —  a^P,  surds  of  the  form  Va^  and  fo^  are  equivalent  sards 
of  different  orders. 


264  ELEMENTS  OF  ALGEBRA. 

6.  A  Mixed  Surd  is  the  product  of  a  rational  factor  and  a  surd  factor ;  as, 
a  V6 ;  3  Vb. 

7.  An  Entire  Surd  is  one  in  which  there  is  no  rational  factor  outside  of 
the  radical  sign;   as,    V2;    \'a^;    Vx. 

8.  A  binomial  surd  has  two  terms,  and  involves  one  or  two  surds;  as, 

a  -\-b  Vx]    a  Vx  —  b  yy-      A  compound  surd  or  polynomial  has  two  or  more 

2  3  -  4  - 

terms,    and    involves    one    or    more    surds ;      as,     y2  +  3  4/4  —  5  V3 ; 

a-\-h-  c  +  2Va. 

9.  Quadratic  surds  are  of  most  frequent  occurrence. 


107.  The  methods  for  operating  with  surds  follow  from  an  appli- 
cation of  the  principles  of  Chapter  XVIII.     Thus, 

f  =  V^f .     2  a2  &3  ==  ^(2^2y3y3  ^  ^^^;^9;     j^  general, 

n 

a  =  a^  =  a^—  ^a^.     Hence, 

I.  To  Reduce  a  Rational  Factor  to  the  Form  of  a  Surd  of 
any  Order.  Raise  it  to  the  power  indicated  by  the  root  index,  and 
place  it  under  the  radical  sign. 

2V'3  =  V2'  X  3  =  Vl^.     f  ^9  r=  ^(1)3  X  9  =  ^|.     In  general, 

n     \  1 

a  ^x  =  an  xn  —  (a'*;r)»  =  y'a^.     Hence, 

II.  To  Change  a  Mixed  Surd  to  the  Form  of  an  Entire  Surd. 

Reduce  the  rational  factor  to  the  form  of  the  surd,  multiply  by  the 
surd  factor,  and  place  the  product  under  the  radical  sign. 

V72  =  V62  X  2  =  6  V2.     ^1029  a*  =  (7^a^  X  Says  =  7  a  ^3a. 

9    3/7   _    9  i3/iZi   _   ?Jl1  ,3/T  ^.VJI    _    «,V     1    X^'« 

^Vu-^y2x4~       2      -  V4.       2V2a3  -  2V2a3  X  23a 

a    /  Sa  ^j — 

"^  2  V  2^  =  i  V 8  a-       In  general , 

Hence, 


RADICAL  EXPRESSIONS.  265 

III.  To  Reduce  a  Snrd  to  its  Simplest  Form,  ii  tiie  surd  is 
integral,  remove  from  under  the  radical  sij^n  all  factors  of  which  the 
indicated  root  can  be  exactly  obtained. 

If  the  surd  is  fractional,  multiply  its  numerator  and  denominator 
by  such  expression  that  the  indicated  root  of  the  denominator  can  be 
exactly  obtaiued. 


\/2^  a  X  v^o^  =  \^'2^a  x  a^  =  a  \/2.     In  general, 

_  —  —  A  J  *  *  * 

^a  X  ^/b  X  ^c  X  . . . .  ^  =  a'*  X  b"  X  C^  X  . . . . p^  =  (abc  . . . . py 
=  \^a  be p.     Hence, 

IV.  To  Find  the  Product  of  Two  or  More  Surds  of  the  Same 
Order.  Take  the  product  of  the  expressions  under  the  radical  signs* 
and  retain  the  root  index. 

In  general, 

^/^-^\/r=(^y=\/'f-    Hence, 

V.  To  Find  the  Quotient  of  Two  Surds  of  the  Same  Order. 
Take  the  quotient  of  the  expressions  under  the  radical  signs  and 
retain  thn  root  index. 

f/i5^64  =  ^64  =  2.      (/ V25^  =  1^(2»)''  =  2^  =  4.     In  general, 

^'^  =  (""j*  =  a^  =  "^a.     Hence, 

VI.  To  Find  the  vith.  Root  of  the  lith  Root  of  an  Expres- 
sion.    Take  the  mnth  root  of  the  expression. 

Note.  It  is  sometimes  easier  to  j>erfonn  operations  with  .simls  if  the  arith- 
metical numbers  contained  in  the  surds  be  expressed  in  their  prime  factorSf  and 
fractional  exponents  be  used  instead  of  radical  signs, 


266  ELEMENTS  OF  ALGEBRA. 

Exercise  94. 

Express  in  the  form  of  surds  of  the  3d  and  nth  orders, 
respectively  : 

1.  1;    |;    22;    4";    2  a";    Sahc;    S  x;    a^;    of;    af'y\ 
Express  as  entire  surds  : 

2.  JV2;   1^3;   5  V32  ;   f  V^;   leVflf;   abVbi. 

3.  a  4/d^ b  6-8 ;    3  a^  ^ofc^  ^    i '^^ ;    2x</J^;   |  ^. 

5.    5-;.^^25^i;    (,»-l)v/^;    '^^±^\I^^EI . 

'  ^  m  —  1     771  —  71^  m  +  n 

^'  ;r^V-^r^'  ^V^?^'  ^"V^r-;  ^^Vs^' 

Express  in  their  simplest  forms  : 

8.  -^288;    3V150;  •^^^^IIS?;    fV90|;    2  a^W^. 

9.  V3i;     ^Jl|;    ^J^;     ^'1029;  Vf;    V^ ;    ^|- 
10.   <1\    ;J|;    ^=T08^a,     ^3^i;;i5ro,    ^Z'^- 

11.  ^^?n=^^\  ^'V^;  !^v/— • 


RADICAL  EXPRESSIONS.  267 


.   -^',    ^7290a3-j6m^2.    ^^J^;    ^a^^+Y"- 


12.     ,/ 

^586 


13.  V(a;  +  y)  (a^  -  ^) ;    Va«2  -  8  aa;  4- 16  a. 

14.  ^^yJtlEI^LlIl,     ^1715 ^^-V^ 

Simplify : 

15.  Vl2  X  Vl8  X  V24;    V54  --  Vl)  V^Vlf. 

16.  ^Fex  ^^54x^/128;    [v0[28^ -h  ^5^6^]  ^  >^9^. 


17.  ^v^^:^.-  X 


V50a8  66^  V32a63 


18.  V2^  a8  ^6  X  vOUe  a2  m2 a:^  X  v^56  a^  m^  x\ 

19.  (^53a«fe9  -4-  -^25  a*  6^)  x  ^125  aH  X  <^W^. 

20.  (V6M  -^  V63~?)  X  v^54^  -T-  v/feT:    \/'^^^^^. 

21.  (^iiT?^  X  ^a-ift-ic)  -^'(v^a:-^oyo  x  ^^lO^X 

22.  (^|^^)^V20736;    'i^ivF^  -  aX^. 


23.  Vf  a8  X  Vf  a-2  X  V.f  ai  X  V2.5  a"*. 

24.  \1\J  </W^^^\    (16  aH2)i  X  (ai  h^f  -r-  ^2  J  h. 

108.    ^5/2  =  •^^21^  =^8.      3^=3*^]|^2^M?  =  3]!J^l6. 

In  general, 

p  pxw  

y^aP  =  «"  (n  >  /))  =  a*  »<  "•  =  "^aP"*.     Hence, 


268  ELEMENTS  OF  ALGEBRA. 

I.  To  Reduce  a  Surd,  in  its  Simplest  Form,  to  an  Equivalent 
Surd  of  a  Different  Order.  Divide  the  required  root  index  by  the 
root  index  of  the  surd,  and  multiply  the  power  and  root  index  by  the 
quotient. 


TheL.C.M.  oftheroot 
indices  (3,  9,  6)  is  18. 
In  general, 


pm 
Pin  

^6w  =  fe"*"  (m  >  pi)  =:  "^6p.».     Hence, 


II.   To  Reduce  Surds,  in  their  Simplest  Forms,  to  Equiva- 
lent Surds  of  the  Same  Lowest  Order.      Divide  the  L.  C.  M.  of 

the  indices  by  each  index  in  succession.  Multiply  the  power  and 
root  index  of  the  first  surd  by  the  first  quotient,  of  the  second  surd 
by  the  second  quotient,  and  so  on. 


Exercise  96. 

Express  as  surds  of  the  12th  order: 

1.    A^2;    ^3;    f^;    3^2;    ^a^;     ^1;    i^^S. 
^     2.    a/sS;    1^32;     ^a^;     v'^^X  V^^^i"^. 

Express  as  surds  of  the  7ith  order,  with  positive  expo- 
nents : 

3.    ^x^;    V^;    ah;    ^'^j};    -L;    v/«~";    ^. 


RADICAL  EXPRESSIONS.  269 

Reduce  the  following  to  equivalent  surds  of  the  same 
lowest  order: 

4    V5,  ^11,  4^;    a/2,  \^5,  \/3;    \^8,  V3,  ^6. 
5.    ^2,  ^8,  ^i;    v^7,  ^5,  ^6;    Va,    ^a^ 
G.    '^^,  Va;    ^^«,  ^a6,  ^a^    ^1^?,   '^^^. 

8.  Vaic^,  ^/a^Q^\    ^fm,  "^n,  v^,  ^mnx. 

9.  v"^,  \^6^  ^;?;    41^5^,  2^VlW^,  10  a  a/37. 

109.    fV6=A/(IF^^=A/i   =a/W. 

i a/5  =  a/(|)''  X  5  =  V¥  =  a/H      .-.  IV'5>Ia/6- 

In  general, 

_  J  

a  J^x  —  (a"  x)"         =  y^a*  as, 

5  ^'y  =  (6-  y)*         =  .y^Fy.      Hence, 

I.   To  Compare  Surds  of  the  Same  Order.     Reduce  them  to 
entire  surds,  and  couipaie  the  resulting  surd  factors. 

\  ^52"=  ^{\y  X  2«  X  13  =  y  ^'  =^^/42:25, 

I  ^8    =  ^{\y  X  2  =  ^  (f)«  X  22  =  ^45.5625, 

3  Vl    =  a/3*  X  f  =   '^3«  X  (fli»  =  >^46.656. 

Therefore,  the  order  of  magnitude  is  3  ^\,  \  ^,  \  >^52. 
In  general, 

—        ??J?    JL 
6  y^y  =  6»»»  ir ■  =  "v^b^-y*.     Hence, 


270  ELEMENTS  OF  ALGEBRA. 

II.   To  Compare  Surds  of  Different  Orders.     Reduce  them  to 
entire  surds  of  the  same  order,  and  compare  the  resulting  surd  factors. 

Exercise  96. 

Which  is  the  greater  ? 

1.  3  V6  or  2  Vl4 ;    6  Vll  or  5  VlSf  ;  4  VG  or  6  Vi 

2.  10V5or4V3l;    iVTorfVlO;    ^^2  or  ^3. 

3.  V|or^l|;    ^4  or  ^5;    Vf  or  ^T|. 

4.  ^11  or  '^f;    1.6  or  J  ylO;    \^6^  or  V^. 
Arrange  in  order  of  magnitude : 

5.  V3,  </4,  </7 ;    8  V2,  5  a/5,  4  V7|. 

6.  2'^2l,  3^49,  4V7;    3^4,  4  ^I|,  2^131. 

7.  3 '^2,  3V2,  |A^4;    2^21,  3^8,  2V8. 
Show  that  the  following  are  similar  surds : 

8.  ViO,  V90,  Vf ;    J  V'20,  i  V45,  5  Vf. 

9.  7  V|,  'V/ff ,  3  VS  ;    -^162,  3  ^32,  </2E92. 

10.    V27,  Vr92,  Vl47,  Vl;  a^W^^  h</W^^  f  V— . 
11     QiV*^    ^V5T2    ^K^l^-    i/«^    .V«*^'^     ^.la^(?m^ 


RADICAL  EXPRESSIONS.  271 

110.   Addition  and  Subtraction  of  Surds. 


iVH"^=-iV^|x2  =  -fV2. 


Adding, 


3»a«Xa^arx(26)g      3  a  o/-—— 
26X726?  =26^^^  ^  ^' 


3/27a«x        s/ 

-°V  26-  -y-2hxm  =  ~  26 ^^^«'*'^' 

1   8/46^       13/4  62a:  X  a*  1   3/ — sttt- 

6V^  =  ^V^=^3r^^  =^lJ/4a262x. 


Adding, 

=  ^^-y-  ^4a^b^x.      Hence, 

Reduce  each  surd  to  its  simplest  form.  Prefix  the  sum  or  differ- 
ence of  the  rational  factors  to  the  common  surd  factor  of  the  similar 
surds.     Connect  dissimilar  surds  by  their  signs. 

Exercise  97. 

Simplify : 

1.  3  Vis  -  2  \/20  +  3  V5  ;    3  \/|  +  2  V^. 

2.  2V|  +  3Vi;    2^l62-J^^^;    3  V^  +  ^?^. 


272  ELEMENTS  OF  ALGEBRA. 

3.  A/3  +  Vli-2V5i;    5v^^=^54-2v/::r6  +  -i'^685. 

5.  S^J^+^Ml-W^;  3\/l62-7^!^32  +  ^1250. 

•    6.  ^?+  1--^^- 3  V^27^2.  ^40-3  v''320  + 4^^135. 


7.    V50ab^c^-VS2aH^- (4:hc^-3ac)V2al)c. 


8.    a;  Vwi^^  71^"^  x^  —  m  vm^  n^^  x^^  +  n  -^m'^  n^  aP. 


9.    V3  a  Z^'^  +  6  a&  +  3  a  +  a/3  a  &2  _  (3  ^  ^  +  3  ^^ 

10.  y/^+Y/^_2«  +  «^-^V^^^. 

^  a  —  0        ^  a  -{-  0  a^  —  h^ 

11-   |^A  +  0-SVf-iVV96+1.5^|-ii^T750  +  8V|. 

111.   Multiplication  of  Surds. 


3  -v/a  X  7  V^  =  21  y4  X  3  =  42  y'3. 

/^2  X  ^3  -  lJ/2^"xT3  =  :^432. 

f  V2  X  1^3  X  lA^  X  -^i  =  f  X  I  X  f  ^£6  X  3^  X  {\f  X  (i)« 

=  <{^28=  -^2.     In  general, 

a  ^x  X  h  y^y  =  a  x""  X  h y^  =  ah  {x  ?/)«  =  a  6  -y^ari/. 


a  yar  X  h 'J^y  =  ax""  X  b  y'^  =  a  &(»;'"?/")"'«  =:  a  &  y'a;"'  i/**.     Hence, 


RADICAL  EXPRESSIONS.  273 

I.  To  Find  the  Product  of  two  or  more  Monomials.  Reduce 
the  surds  to  the  same  order  (if  necessary).  Prefix  the  product  of  the 
rational  factoi-s  to  the  product  of  the  surd  factors. 


Multiplicand, 
Multiplier, 

3V3, 
2^6, 

3V2- 
3V3H 
9^6- 

9V«H 

-2>^5 
h2^ 

3  >y/2-2v^5  multiplied  by 
3/y/2-2'v^5  multiplied  by 

Sum  of  partial  products. 
Hence, 

-6^5<^X3» 

6  ^2«X  62-4/^3(7 
h  6  v^288  -  6 -^'675-4  v^30. 

II.  To  Find  the  Product  of  two  Polynomials.     Proceed  as  in 
Art.  24. 

(^^/2-\-2^3){^^2-2^/3)  =  (3  X  2*)^-(2  X  3*)^^  =  32  X  2-2^  X  3  =  6. 
(a^x-^b\/~y)  (a^x-bx/y)  =  (a x^^  -  (hy^f  =  a^x-h^y.     Hence, 

III.  The  product  of  the  sum  and   difference   of  two   binomial 
quadratic  surds  is  a  rational  expression. 


Exercise  98. 
Simplify : 

1.   2'v/r^  X  3  V3;  SVf  X  J\/T62;  J  VlO  x  J^  Vl2j. 
2\/l4  X  V2i;    3^1  X  gVJ. 
(5  V3-5)  X  2  V3;    \!^64x2V2- 

4.  (V2  +  V3  +  2  \/5)  X  V2 ;   4  ^75  X  2  V^. 

5.  J  V4  X  v^iO;    i  VJ  X  §  V^l;    V5  X  ^2. 


2.  J  ^4  X  3  v^2 

3.  3  >^3  X  3  V2 


6.    3  \/|  X  ^1  ;    J  \/§  X  9  -^1  X  \^. 

18 


274  ELEMENTS   OF  ALGEBRA. 

7.  2^3  X'v/2  X  J'^i;    V^%X</'i;    ^T68  x '^147. 

8.  '^^X^9X^9*;  (3V2-3'v/6-V8+3V'20)x3V2. 

9.  a/5  X  V^IO  ;    (Vn  -  Vm)  xVn;    4  \/^ x  3  VS. 

10.  Vmn  X  \^S  m^x  X  V2  nx. 

11.  "^/m^no^  X  "^m^  n  x ;    2  Va  X  '^^  X  3  ^a  x  '^^. 

12.  ^^2^  X  ^3^  X  V -i-;    ^^(4  7/^  a^'^)"  X  ^(:2m^x)\ 

13.  V-xtV^r^;    (V2-3V3)(2V3  +  3V2). 

71   ▼    71  3   ▼    2  ft* 

14    (3V5-4^2)(2^5  +  3V2);    (^2  +  ^3)^. 

15.  (5  V3  -  6  a/2  +  a/5)  (lO  a/3  +  12  a/2  -  2  a/5). 

16.  (V2  +  </l+   '^D  «/2  -  V3);    '^24  X  6  ^3. 

17.  (V2  +  V3)  (V2  -  VS);    (^3  +  ^4) (^3  -  m 

18.  (a/5-a/3)(V5  +  a/3);    (a/5  +  2  a/3)  (a/5  -  2  a/3). 

19.  y^l2  +  a/19  X  y^l2- a/19;    v'TG  X  a/S. 

20.  y'9  +  A/n  X  \^9  -  A/rZ ;    a/3  X  ^2  X  ^|. 

21.  (^a3  +  ^^)  (^-2  _  ^-3) .     y|  ^  ^1 


RADICAL  EXPRESSIONS.  275 


22.  V^10+  V68  X  y^lO- V68;    -v^^^xy^^. 

23.  {dVx+3  Va-^x)  (5  Vi  -  3  Va-2^x), 

24.  ^S  X  ^M;  VW  X  ^i;  {mfi<^Mf' 

112.  To  Rationalize  Surd  Denominators  of  Fractions. 

2                  2  X  V3      2  V3       2 
;^  = 7^^ -7-  =  -^  =  T^  X  1.732  +  =  .23  +  . 

2  2X  -v/3^«      2  ^3«      2  ,/_ 


— =r=  =     - — !^, =  — -  —    (n>m)  =  — T .    Hence, 

I.   If  the  Fraction  be  of  the  Form  — — = .     Multiply  both 
terms  by  y"^^^^-  ^  ^^ 

3+  V5  _  (3  +  ys)  X  (3  +  ys)  _  3«  +  2  (3)  ( /y/s)  +  (^5)* 
3  -  V5  ~  (3  -  V5)  X  (3  +  Vs)  ~  32  -  ( VS)^ 

14  +  «V5      7  +  3X2.236+      ^    ^^ 
=       9-5       =  2 =  ^•®^^"^- 

4V3  +  3a/5       (4  a/3 +  3^5)  X  (2V7-3  V2) 

2  V7  +  3  >v/2  ~  (2  V7  +  3  V2)  X  (2  a/7  -  3  v^ 

_  8  \/2l  +  6  a/35  -  12^/6  -  9  a/To 

g ox(A/^TA/g)  _  aiVbTVc)  _  a{\/bT\^c) 

V^±A/^"(A/^iA/^)x(A/ftTA/^)"  W~-W~     ^"^ 

g a  X  Tfe  T  A/g)  _  ajh^^c)  _  a(hT  Vd       „ 

b±^~  (b±\^c)x(bTV~c)~  (py-ic^^  "      ^*  ~  *^  ^"^' 


276  ELEMENTS   OF  ALGEBRA. 

II.  If  the  Denominator  is  a  Binomial  Involving  only 
Quadratic  Surds.  Multiply  both  terms  of  the  fraction  by  the 
terras  of  the  denominator  with  a  different  sign  between  them. 

Note.  It  is  often  useful  to  change  a  fraction  which  has  a  surd  in  its  de- 
nominator to  an  equivalent  one  with  a  surd  in  its  numerator.    Thus, 

8  SXVI      8  I'S^  J  X  2.236+=  1,3416+. 


V5        V5  X   V5         5 

Exercise  99. 

Eationalize  the  denominators  of: 

2  3  2  -  ^2  3  V5 


1. 


V2  +  V3'    2  V5  -  V6      1  +  V2      V3  +  V2 

8-5  V2,    2  V"5  -  V2  1  6 

3- 2  a/2'    V5  +  3V'2'    3-2  Vg'    'V^64  ' 


Vx  —  Vy ,    Sx  —  Vx  y  _     Va  +  a;  +  V<?'  —  x 
o. 


"s/x  +  Vy      Va:  y  —  2>y      Vet  -\-  x  —  Va 


X 


.    X  —  Vx^  —  1  a  1  2  a 

4. 


+  Vx^-l'     \/a+Vb      V5-V'2'    3a/2^-^ 

Given    V2  =  1.414,    V3  =  1.732,    V5  =  2.236 ;    find 
the  approximate  values  of: 

5.  ^_;    V50;   8K288     ''  '  ' 


6. 


V2  ^  V5     2V675      V500 

1  +  V2    1-V5  3  1.1 


2  +  ^/2'  3+V5'  21/2-3^/3    ^5-^2    2  +  ^3 


RADICAL   EXPUESSIONS.  277 

113.   Division  of  Snrds. 

2  V54 -^  3  ^6  =  I  VV     =  f  X  3      =2. 

Ingeneral,     a^i^6^y  =  ^^)"      =^Vy* 

I.  If  the  Divisor  is  a  Monomial  Reduce  the  surds  to  the 
same  order  (if  necessary).  Prefix  the  quotient  of  the  rational  factors 
to  the  quotient  of  the  surd  factors. 

,-     .      ,-  ,-x  3\/3  3  V3X  (3^3-2^2) 

3  V3  -r  (3\/3  +  2  V2)  = :^ p  =  7 7= V\    /      y         A 

^  ^  ^  3'v/34-2V2       (3V3+2-v/2)x(3V3-2V2) 

^27-6V6      Hence,  in  general, 

II.  If  the  Divisor  is  a  Binomial  Involving  only  Quadratic 
Snrds.  Express  the  quotient  in  the  form  of  a  fraction,  and  ration- 
alize its  denominator. 

^a -^  \/b  -  \/c) a  +  2  \/ab  +  b-c  {\^a+  \/b-\-^c. 

Divisor  multiplied  by  '\/a,  a  +     \/a  b  —  \/a  c 

First  remainder,  \/a  6  +  6  +  V**  c  ~  c 

Divisor  multiplied  by  y'ft,  \/a b  +  b  —  ^/bc 

Second  remainder,  y'a  c  +  ^b  c  —  c 

Divisor  multiplied  by  \/cj  ^ac  +  \/6 c  —  c 
Hence,  in  genenil, 

III.  To  Divide  a  Polynomial  by  a  Polynomial.  Proceed  as 
in  Art.  33. 


278 


ELEMENTS  OF  ALGEBRA. 


Exercise  100. 

Simplify : 

1.  21V384^8V98;    5  \/27  ^  3  V24;    \/l2^V^24. 

2.  -  13  VT25  -^  5  V65 ;    6  Vl4  H-  2  ^21. 

2V98   '    7  a/22'    5V112   '    V394 '    ^2   *   Vs' 
4    IJ  ^2|  -  I  Vll;    -^12  --  ^2 ;    V6  --  A^4. 
5.    20  ^^200  -4-  4  a/2  ;    ^18  ^  a/6  ;    4  ^32  ^  '^IG. 

3  a/108       5  a/14       15  a/84 

7.  (15  a/105  -  36  v'lOO  +  30  A^81)  -^  3  Vl5, 

8.  '^OOei^ViO;    a^'a^c-^^^;    Va -^  \^. 

m  —  71  ^  m  —  n       ^  {m  ~  nf         '3  »2 

10.  {acx^Vy—hcy  ^/x)  -^  c  a/^  ;  '\/a~x  -^  ^'o^. 

11.  ^4  m  7^2  -H  V2w3^;    v^2W^  X  ^^^?^3^  aA;?^5 

12.  A^4^i2^XA^'9^^i2^*^v'25^^2^;    <^d~^-r-\/^' 

13.  -y— ^x-V/-2--^— -V/-|— •   (aj-l)-f-(A/aj-l). 


RADICAL  EXPRESSIONS. 


279 


14.  V10.4976  -^  2  Vo  ;    (2  a:  -  Vo;  y)  ^  (2  Va:  y  -  y). 

15.  (:^  a/3  +  2  V2)  ^  (a/3  +  V2)  ;    4  ^a^  -f-  3  Vo^. 
,^    2A/T5  +  8  .   8V3+  6a/5     8-4\/5  .  3a/5-7 

Id.      =:r-  -7-   — —',      7^7- -T- pr-  • 

5-  a/15        5a/3-3a/5      1  +  a/5       5  +  a/7 


17.  (^x«  +  ^^v  +  w")  -  (V'^'^"  -  va;V  +  vy). 


114.   Involution  and  Evolution  of  Surds. 

l^v/ir=[Mi)T=i-.x®'=4v/I=^vi.- 

y486av/4a«  =  [3«  X  20(220*)*]'  =  [3^  X  2^a^f  =  3  X  2*0*  =  3^/2^ 

m  mp 

In  general,     (a^i  -v^t^)^  =  (a«i  6"*)'  =  a"!^  6^  =  a^h^  >v/6*^. 
Vo'»i  v^6™  =  (a"*!  6"  j^  =  a  »•  6"  ^     Hence, 

Express  the  surd  factors  with  fractional  exponents,  and  proceed  as 
in  Art.  104. 

Example  1. 
V  A«/;5        2a)    ~  Ul      2aJ 

-©■-3Gr(i).3@C4)'-e)' 
-^  -'S)©"K)(^)- 


^? 


2»a« 


3a* 

2cH 
3q« 


3 


4a*c* 
3 


i«ra 


8a» 
8a«" 


280 


ELEMENTS  OF  ALGEBRA. 


'i 

CO 

+ 


I 

+ 

+ 

-0 

Li 


i<5 

^ 

CO 
1 

1 

1  y 

1 

'> 

1 

^ 

0 

+ 

+ 

-"« 

Ol 

CM 

II 

Th 

+ 

Hw 

-* 

« 

c 

Tt^ 

^ 

I 


HM 

-O 

CO 

1 

^ 

% 

GO 

GO 

+ 

+ 

Hn 

c 

c 

-^ 

Tf 

'-Ca 

^5 

a 

r^ 

'd 

S 

S 

c 

^ 
0 

1 

fn'' 

fi" 

.2 

-U 

jT 

p-T 
a; 

J 

^h" 

'IS 

ir^ 

-M 

c 

0 

s 
s 

^ 

(V 
S 

id 

1 

S 
0 
0 

1 

i 

0 

s 

s 

-tJ 

C 
0 

8 

-^ 

^ 

-TIJ 

-n 

-s 

02 

CK 

03 

^ 

rt 

a 

G 
0 

rt 

^ 

^ 

ti 

^ 

£ 

0 

•>-( 

o; 

S 

£ 

s 

pR 

pR 

'/2 

m 

02 

02 

RADICAL  EXPRESSIONS.  281 

Exercise  101. 
Find  the  values  of  the  following : 
1.   mf;    ^VE;    i^Vlf;    ^^2;    (^32)'. 


2.  'fe^;    ^m-,    i'^I^jf;    V-^Gi. 

3.  {</uf;  vQ^;  mf;  ^vff;  mt 

4    ^JV^;    (^27)^    --(^A;    (2^3¥f ;    'sp^- 


5.  (2  ^^6)';    y'V©"'    '^"•^^■^"^    (2a^2Fo/. 

6.  ~^/•i-•a-'■,    {Z'^Wc^f;    ^iU^";    "v''27»=^. 

7.  'v'»-»v'^»;     [\/(a-c/]";   "-^1;    [jv^I^^-^js, 

11.  v^9-a:— ;     [{x  ^  y)  V^y^  -     VnS^2i^^, 

Find  the  values  of  the  following,  and  express  the  results 
in  terms  of  positive  exponents,  by  inspection  : 

12.  (^^^T?f ;    Wl  +  V\f;    {V2  +  Vsf. 


282 


ELEMENTS   OF  ALGEBRA. 


13.    (^3^l)MV^+^i)-     ["^-fl- 

15.  M^.o]^  [(-^«--f +(fr^)-T- 


16. 


[jn^  J'Zm       2-^mT      ["^    /m-^       4 -^^3a"[ 
12  mV    n    "^    4/n  J  '     L    V    ^4         ^^^^^3  J 


Find  the  square  roots  of : 

17.  ^  +  1  +  9  ^^  +  1^-1  _  1^-1  ^^  _  6  ^^. 

18.  1  -  2"+^  +  4";    9"  —  2"+^  X  3"  +  4". 


19. 


r 


V2. 


ic       4?/  -^a;  '2/ 

m  

20.    v"^  -  4''^2;5-  +  4  :i:"^  +  2  a/^^ "'  -  4  '-v^^^ '"  +  a/^' 


Miscellaneous  Exercise  102. 

Find  the  values  of  the  following  : 

1.    'V^;     v^;    V3xV27;    1"";    "'^4;   a/X^V|. 


2.'    ^^32^;    V^e^m^^     V^m  ^  V^ 
Reduce  the  following  to  their  simplest  forms : 


18|'    -^'2'- 


3.  t  J^,;    V^;    ^3888;    ^!^±^sj-^ 


») 


RADICAL  EXPRESSIONS.  283 

Reduce  to  equivalent  surds  of  the  same  lowest  order : 

5.  V2,  '^,  •^5;    -v^y,  a!^8,  4/5;    3^75,  3v^54. 

6.  2A/I8,  ?,\/U,  4A!^r62,  5-^128;   V^\  </V,  i^^. 
Change  to  similar  surds : 

7.  •^27,  -^144;    -C^Si,  3^;    W2,  ■^243. 

8.  ^02,  VI;    VTo,  4'Wb;    I'^lh,  V^. 

9.  2\/^.    ^7-26^,    ly^^;     K.1G;  4^275. 

10.  n,  VJl,  ^i^,  3  ^^;    V20,  3  '^.  4  Vl25. 

11.  ^32.    ^128;  VM^.    y/^'.   SfW^'- 

12.  ^J^.  ^192,  ^S".  ^^.  ^A;    ^2,  ^2?. 
Arrange  in  order  of  magnitude  : 

13.  3  a/2^,  4  </^,  3  -^3 ;    5  ^8,  3  ^9,  3  VlO. 
14   fnfV3~,  iV5,  2^;    Vi    ^H- 

15.  a'  -e^^sTjy,  -^  ^(25^^-,  (64)"  sj"^, . 

Find  the  values  of: 

16.  V243  4-  \/27  4-  v'48;    2  V^189  +  S'l^'STS  -7^, 


284  ELEMENTS  OF  ALGEBRA. 

17.  4  V5  X  'V^lT;  3  'v'^gOO -^  \/5;  \J2  V  2Vl  -^  ^V^^' 

18.  5V'2  +  3V8-2V32;     3 -^^ST  -  4 ^1^192  +  ^648. 

19.  'V^m  X  #432 ;    I  V5  X  |-  #2  x  #80  x  #5. 

20.  ^64+5-^32-^^108;    i(^27  +  | -V^i92  +  •^81). 

21.  2'v^JJ^  +  a/60  -  \/225  -  Vf ;  J  V|^ { a/2  +  3 V| )• 

22.  (a^|^2^^)--V^I6;    (-^9-2^21  +  4^1)2-^9. 

23.  (6  -^1  +  ^18)  -f-  -^72  ;    1|  -^/20  -  3  -v^5  -  -v/f 

24.  J,a  ^60-^(2  ^240  +  7^31);    y/j^-^^^)*. 

25.  (-^ro  -2-^4  +  4  -^54)  (o  -^64  +  sV^-  2  ^32). 
Eationalize  tffe  denominators  of: 

^g    V2Q  -  a/8.    (3+  V3)(3  4-  #5)  (#5 -2), 
*    V5  +  V2  '  (5  -  V5)  (1  +  VS) 

^    m  -\-  (171  —  Vvi^  +  a?  71^     x"'  2 


m  —  a  ?l  +   V??i2  ^-  ^2  7^2  „  '      ^5  _|_   /y/3  _  ^2 

Simplify  the  following : 

28.  ^_        ;  V(f|y"  X  V(||f ;  #(8'a3&)2  x  #(2  a &3)2. 

V2 

^^    7  +  3  \/5  _^_  7-3a/5^        /^9?»Ti  x  V3l<T" 
^"   7-3^5      7  +  3A/5'    V  3V3^ 


30 


.  {a  -r  #«)"  +  '  y^(^  j/^"")"';     #^2^^""*  X  \x^yl 


RADICAL  EXPRESSIONS.  285 


31.    4^aX'i^:r^X  ^a*  X    "/a  X  ^aV  x  ^^. 


33.  4.x^<77xf  x-^^^„-^(^y-^^/*^ 

34.      V(a>)   ^  v/(-^.,p;    1^,;    |(|)t. 

35.  .^-  (g)  ^  Vl^. ;  I  ^1  + 1  ^^'  -  2  (S)«. 

37    V^  +  «^         V^  ^^  ^  4  ^^ 

38.  ^-Hl^^^i -H  f  1  +  -^V;    i^a^  +  ^a)3. 

^5  X  ^^3 
40.  Express r^ with  a  single  radical  sign. 


Queries.  What  sign  is  given  to  the  Titli  ])ower  ?  To  the  nth 
root?  Why?  How  change  the  order  of  a  suixl  ?  In  T.,  Art.  112, 
why  take  m  less  than  n  ?  How  rationalize  a  sunl  denominator  ?  What 
powers  of  n^ative  nuniWrs  are  positive  ?     What  n^ative  ? 


286 


ELEMENTS  OF  ALGEBRA, 


Imaginary  Expressions. 

115.  An  Imaginary  Expression  is  an  indicated  even  root 
of  a  negative  expression  ;  as,  V—  a  ;  a  -\-h  V—  1.  V—  1 
is  an  imaginary  square  root ;  a  V—  1  is  an  imaginary 
fourth  root;   etc. 

-V^T^^  :^  ^a2  X  (-  1)  =  V«^  X  V^  =  «  a/^^- 
/^-TJ  -  .^6  X  (-  1)  -  ^b  X  V-^-     Hence, 
Every  imaginary  square  root  can  he  expressed  as  the  product  of  a 
rational  or  surd  factor  multiplied  by  \/—  1. 

The  successive  powers  of  ^y/—  1  are  found  as  follows  -. 

)ip=(-l)i  =  +  V=T; 
)*/=(-!)    =-l; 

)*r  =  (-i)'  =  (-i)(-i)*  =  -V~; 

)J]'=(-l)^  =  +  l; 

)i]«=(.-l)»  =  -l; 

)i]»=(-l)^  =  +  l; 

)J]»  =  (-  1)1  =  (-  i)^(-  i)i  =  +  ^—;    and  so 

)*]'=(- 1)5  =VPT)=       -±-v/^orTl, 
oc?c?  or  eyen  integer.     Hence, 
The  successive  powers  of  \/~  1  form  the  repeating  series  : 

+  V~h  -h  -V^»  +1- 

The  methods  for  operating  with  imaginary  expressions  are  the 
same  as  those  for  surds ;  but  before  applying  the  methods  it  is  better 
to  remove  the  factor  /y/—  1.  All  cases  of  multiplication  can  be  made 
a  direct  application  of  Arts.  97,  114. 


w- 

1?  = 

=  [(-1 

w- 

-xY- 

=  [(-1 

w- 

~xY-. 

=  [(-1 

w- 

-lY- 

=  [(-1 

w- 

-^Y- 

=  [(-1 

[V- 

-xY-- 

=  [(-1 

[V- 

1]'  = 

-  [(-1 

w- 

T]'  = 

=  [(-1 

w- 

T/  = 

=  L(-i 

on.     In 

general, 

W~ 

T]"  = 

=[(-1 

accordin 

gas 

n  is  an 

RADICAL  EXPRESSIONS.  287 


niustratioiiB.     ^/-  6aH^=  y/S  a^b*  X  (-1)  =  \/S  aH^  X  \/-l 
=  2ab  \/Tb  X  V^. 


V-9«^  +  V-49a«-V4a'«=  3a  ^-  1  +7a/v/-  1  -2  a 
=  10a  V^- 2a 
=  2  a  (5  V=n  -  1). 

3  >v/^  X  4  -v^^  =  (3  V^  X  \/-^)("*  V^  X  V-^) 
=  3  ^3  X  4  y  2  X  ^/^  X  V^ 
=  12V3X^X  [(-l)*]' 

=  - 12  ye. 

2  -v/^  X  5/v/^  X  3  V^ 
=  2  V3  X  5  V2  X  3  >v/6  X  \/^  X  ^^^  X  V"^ 
=  30>v/3  X  2  X  6  X  [(-  l)*]" 
=  -  180  ^~l. 

=  f  -v/3  X  1  =  J  V^- 

_  (i + y-H)'  _  i+.2v^_+ (-0 

Example.     Multiply  1-2  V"^  by  3  +  ^y/^. 
ProcesB.  1—2  ^—~i 

3+     V-1^ 


1  -  2  -y/-  1  multiplied  by  3,  3-6  V"  ^ 

1-2  y'lH!  multiplied  by  y'^,  2  +     V^ 


Sum  of  the  partial  products,  6  —  5  /y/—  I 


288  ELEMENTS  OP  ALGEBRA. 

Notes :  1.  Imaginary  expressions  represent  impossible'operations ;  yet  it  is 
a  mistalie  to  suppose  that  they  are  unreal,  or  that  they  have  no  importance. 

2,  If  the  student  employ  the  method  of  multiplying  or  dividing  the  expres- 
sions under  the  radicals  (Arts.  Ill,  113),  for  all  cases  in  multiplication  and  divi- 
sion, he  cannot  readily  determine  the  sign  of  the  product  or  dividend.     Thus, 


V^^  xV—a  =  V—aX—a=  Va^  =  ±a. 

3.  Is  the  above  product  both  ±a  or  —  a  ?  We  are  limited  to  the  considera- 
tion of  the  product  of  two  equal  factors,  and  we  know  that  the  sign  of  each  is 
negative ;  also,  that  Va^  =  it «.  Hence,  the  sign  of  Va^  will  necessarily  be 
the  same  as  that  of  each  of  these  factors.    Therefore,  it  will  be  the  same  as  was 

its  root.     Thus,  

V-  3  X  1/-  3  =  -  1/9  =  -  3, 


Exercise  103. 

Simplify : 

1.  V^;    '^-16;    V- 12  a;     V^^T^;    V^ 

2.  V-49a2-&6.    ^^7729;    ^IT^;    y'^^^". 
Find  the  values  of : 

3.  (V— i)i^,(V^f ;   iV-if;   {-V—lf. 

4.  i-V^lf;    (-V=^r;   i-V=-lf;  {-V^f 


5.    A/-25  -  A/-49  +  V-121  -  a/-64  +  V-1- a/-36. 


V-22       V-216 


6.    2  V-  24  +  —=  -  V-  18  ;    ^...^  - 


V-3  A/-33       V-324 


7.    V-  36  a^  4-  V-  9  a^  -  V-  (1  -  af  a^  -  V-  a\ 


8.  V-{ct-hf+  V-(a2- 2ab  +  b^)+  V-1 6 a^ b^-V- 4 a^ 
Multiply : 

9.  V^  by  V^;    3  V^  +  V^^  by  4  V^^ 


RADICAL  EXPRESSIONS.  289 

10.  2  V^  by  4  V'^;  1  +  V^  by  1  +  V^. 

11.  V-  2  +  3  V^  by  V^^  +  3  V^. 

12.  3-2  V-4  by  5  +  3  V^^;  4  +  V^  by  4- V^. 

13.  1  +  V^  by  1- V-1;    2  -  V=^  by  1  -  2  V^~3. 

14.  2  V^  -  6  V^  by  V^  +  V^. 


15.  Va  —  ^  by  V^  —  a ;    a  +  V—  a;  by  a  —  V—  a;. 

16.  a  V—  a  +  b  V—  b  by  a  V—  a  —  5  V—  6. 
Divide  : 

17.  V^^  by  V^^;    -  \/^  by  -  6  V^. 

18.  V^  by  V- 20;    V- 24  -  V^  by  a/^^ 


19.  2  V—  4  «*-»  by   V—  a^  \    a  +  V-  a  by  V-  a^. 

20.  -  2  V^  by  1  -  \/^ ;    2  by  1  +  V^. 

21.  \^-^^^  by  v^-  5;    '^^^^  by  ^=^. 

22.  4  +  V^  l)y  2  -  V^;    V^3  by  1  -  V^. 
Rationalize  the  denominators  of : 

23  ^i^J^^^-  2  \/:ri  _  3  yry   3  +  3  V^ 

"  '   2  -  V^'    4  a/^  +  5  V^^'    2-2  V=I ' 

Queries.  To  what  form  can  all  imaginary  monomials  be  reduced  ? 
In  multiplication  and  divi.sion  why  separate  the  imaginary  expres- 
sions into  their  sunl  and  imaginary  factors  ?  Is  it  necessary  in  all 
? 

19 


290  ELEMENTS  OF  ALGEBRA. 

Quadratic  Surds. 

116.    I.  A  quadratic  surd  cannot  equal  the  sum  or  differ- 
ence  of  a  rational  expression  and  a  quadratic  surd. 

Proof.     If  possible,  let  ^/a  =  6  db  ^\fc,  in  which  ^a  and  y^c 
axe  dissimilar  quadratic  surds,  and  6  a  rational  expression. 

Square  both  members,  a  =  6"^  ±  2  &  ^/c  +  c. 

±  a  T  ^^  T  c 


Transpose,        ±  2  6  \/c  =  a  —  ft^  —  c*.     .♦.   ^/c 


26 


That  is,  a  surd  equal  to  a  rational  expression,  which  is  impossible. 
Therefore,  ^\/a  cannot  equal  h  ±  ^ c. 

II.  i/"  a  +  Vb  =  X  +  Vy,  in  which  a  and  x  are  rational 
and  Vb  and  Vy  cire  quadratic  surds,  prove  that  a  =  x  and 
b  =  y. 

Proof.  Transposing,  /y/6  =  (x  —  a)  +  V^?/-  Now  if  a  and  a;  were 
unequal,  we  would  have  a  quadratic  surd  equal  to  the  sum  of  a  ra- 
tional expression  and  a  quadratic  surd,  which,  by  L,  is  impossible. 
Hence,  a  =  x.     Therefore,  ^Jh  =  ^^y,  ot  b  =  y. 

III.  7/"  V  a  +  Vb  =  Vx  +  Vy,  prove  that  y  a  —  Vb 
=  Vx  —  vV- 

Proof.     Square  both  members,  a  +  \/b  =  x  +  2  aJx  y  +  y. 
Therefore  IL,  a  =  x  +  y     {!)  and  a/6  =  2  ^/x^     (2) 

Subtract  (2)  from  (1),         a  -  \/b  =  x  ~  2  \/xy  +  y. 
Extract  the  square  root,  V  a  —  \/b  =  y\/x  —  \/y. 
Similarly  it  may  be  shown  that  if  V  «  —  ^/b  =  ^/x  —  /y/jr, 

then  V  a  +  ^Jb  =  ^Jx  +  ^/y. 


RADICAL  EXPRESSIONS.  291 

Square  Root  of  a  Quadratic  Surd. 
117.   To  find  the  square  root  of  a  binomial  surd  a  ±  Vh. 


Process.     Let  Va  ±  a/6  =  V^  ±  Vi^  (1) 

Then  (III.,  Art.  116),  Va  T  V^  =  V-^  T  \/y  (2) 

Multiply  (1)  and  (2)  together,  ^af^  b  =  x-  y      (3) 

Square  (1),                                    a  ±  aA  =  x  ±2  ^x  y  +  y. 
Therefore  (II.,  Art.  116),  a=x-hy  (4) 

Add  (3)  and  (4),    a  +  v^^"^  =  2 x.     .-.  x  = ^ 

, a  -  Va*  —  b 

Subtract  (3)  from  (4),  a  -  V  a*  -b  =  2y.     .'.  y  = ^ 

Therefore,  V7±Vb  =  \J^^  V^^  ±  \/"  ~  ^"'~^      (0 

HotM;  1.  Evidently,  unless  d^  —  h  be  a  perfect  square,  the  values  of  Vx 
and  Vy  will  be  complex  surds  ;  and  the  expression  Vx  +  \  y  will  not  be  as 
simple  as  V  a  +  1/6. 

2.  Since,  Va'^c  +  Vbc  —  \/c{a  ^-  Vh\  also  if  a^  -  6  be  a  perfect  square 
the  squats  root  of  a  +  Vh  may  be  expressed  in  the  form  Vx  4-  Vy,  the  square 
root  of  Vcflc  ■{■  Vbc'is  of  the  form  \'c  ( V'x  -f  Vy ). 

3.  Frequently  the  square  root  of  a  binomial  surd  may  be  found  by  in- 
si>ection.    Thus, 

FiTid  ttoo  numbers  whose  sum  is  the  rational  term^  and  whose  product  is  the 
square  of  half  the  radical  term.  Connect  the  square  roots  of  these  numbers  by 
the  sign  of  the  radical  term. 

Examples  :    1.     Find  the  square  root  of  3^  -  VlO. 
Process.    Let  Vi  -  Vy  =  "^H  -  aAo       (1) 

Then  (III.,  Art.  116),  y'i  -f  Vy  =  V^T'^       (2) 

Multiply  (1)  and  (2)  together,  x-y=  \/*^  _  lo  =  |     (3) 

Square  (1),  x  -  2  V^  4-  y  =  3i  -  -  /y/lO^ 

Therefore  (II.,  Art.  116),  z  +  y  =  3|  (4) 

From  (.3)  and  (4),  x  =  2|,   y=l. 

Therefore,  ^3^  -  \/Tb  =  a/|  -I  =  ^  V^- 1- 


292  ELEMENTS  OF  ALGEBRA. 

We  may  employ  the  general  form  (i).     Thus  (since  a  =  83  and 
y6  =  +  12  V35), 

L^  ^  ^g^2  _  (|2  ^35)2 

2.    V83  +  12  ^/3b  =  \ ^^ — - 

4/83  -  V8'3^  -  (12  V35f  _      /83  +  43  /83  -  43 

=  ^63  +  ^20  =  3  v^  +  2  Vs. 


3.    Va/ST  -  2  a/6  =  V  V3  (3  -  2  a/2)  rr:  ^3  X  a/3  -  2  a/2,  ^ 


also  \/3  -  2  a/2  (in  which   a  =  3   and  a/&  =  -  2  a/2) 

^  i /3  +  V3-^  -  (2  A/2y  _  i /3  -  V3"^  (2  a/2)^  ^  ^  _  i 


.-.  a/ a/27  -  2  Ve  =  a!^3  (a/2  -  l). 

4.   Find  by  inspection  the  square  root  of  103  —  12  a/h. 

Solution.  The  two  numbers  whose  sum  is  103  and  whose 
product  is  (^ — |— j  ,  are  99  and  4.  Hence,  Vl03  -  12  \/Tl 
=:  a/99  -  a/4  =  3  a/iT  -  2. 


5.    Similarly,  VlO  +  2  a/21  :=  a/7  +  Vs,  because  7  an^  3  are 
the  only  numbers  whose  sum  is  10  and  whose  product  is  (a/21)  •       ^  ^ 


Exercise  104. 

Find  the  square  roots  of: 

1.  7-2A/rO;    5  +  2V6;    41-241/2;    2J  +  V^. 

2.  18-8  V5  ;    11  +  2  VSO  ;    13  -  2  a/42. 

3.  15~V56;    47 -4  a/33;    6-2^5;    10  +  4  a/6. 


RADICAL  EXPRESSIONS.  293 


5.  V27  +  a/15  ;    2?7i  +  1  +  2  Vm^  +  7i  -  2. 

6.  (m^  +  m)  ?i  -  2  vi  n  Vm ;    9  —  2  VU. 

7.  (wi  +  w)'-^  —  4  (m  -  /O  A/m?i ;    3  a;  -  2  a:  V2. 

Find  the  fourth  roots  of: 

8.  97-56V3;    |  a/5  +  3J  ;    56  +  24  V5. 

9.  17  +  12V2;    4(31 -8  Via);    248  +  32  V^. 

Simple  Equations  Containing  Surds. 


118.    Examples  :  1.    Solve  V4  ar^  -  7  x  +  1  =  2  a;  -  4        (1) 

Process.     Square  (1),  4x^-7x+l  =  4x^-7\x  +  ^. 
.'.  X  =  ]l\.     Hence, 

To  Solve  an  Equation  containing  a  Single  Surd.  Arrange 
the  terina  bo  as  to  have  the  surd  alone  in  one  member,  and  then  raise 
each  member  to  the  power  indicated  by  the  root  index. 

Note.  If  the  equation  contains  two  or  more  surds,  two  or  more  operations 
may  be  necessary  in  order  to  clear  it  of  radicals.    Thus, 


2.  Solve  \/'2b  x-1%-  ^4x-  11=3  -^/i. 

Process.    Transpose, \/25a:-29  =  3  ^/x  4-  *J\x-\\  (1) 

Square  (1),  25x-29  =  9x+6  V(4a:- ll)a:+4a:-ll. 

Transpose,  etc.,      'v/(4a;-  11)  x  =  2  x  —  3  (ii) 

Square(2),  4x«- llz  =  4a;«-12ar  +  9.     .-.  x  =  9. 


294  ELEMENTS  OF  ALGEBRA. 

3.    Solve    ^^-7= =  -^ (I) 

\ X  +  n        ^x  +  3n  ^  ' 

Process.     Clear  (1)  of  fractions,  transpose  and  unite,  etc., 

/           ^     r                 TT              /-        mn                      (  mn  \2 
(m  —  n)  \/x  =  mn.     Hence,  \/x  = .     .*.  x  = 


^  .        \/m  +  X  4-  ym  —  x 

4.    feolve    ^'  ,  J  —  n. 

\m  -\-  X  —  's/m  —  X 

Process.      Rationalize  the  denominator, 

fn.  +  's/m'^  —  x'' 


(1) 


From  (1),  y'rn^  -  x"^  -  nx  -  m  (2) 

Square  (2),  m^  ~  x^  =  n^ x^  -"Imnx  ■\-  m\ 

Transpose,  etc.,  a;^  (l  +  n^)  =r  2  mna:. 

Divide  bv  a;,       a:(l  +  w^)  =:  2mn.      .*.  a:  i=  7— — ;, 

Exercise  106. 

Solve : 


1.  V^  +  5  =  4 ;    V3  ^^  +  6  =  6  ;    ^x^-2^x^% 

2.  Va:2  _  3  ^.  _}_  5  ^  :i;  _  1  ;      ^2^-3  -1=2. 

3.  V'3  + V4  + V^^^=  2;  Vr+W^^m  =  a:  +  2. 

4.  ^/mx^~a  =  'v/c^TT ;     V^rr2  +  ^4  -  ^3^  =  2:. 

5.  VJT~2  =.  2  V'2ic-3  ;    V3  a:  +  5  =  3  ^"Ix  ^  1. 


6.    V3.r- 4=  V2a:  +  16;    ^^2  ^' -  4  =  V 4  -  V2  rr. 


7.  3Vi  = 


RADICAL  EXPRESSIONS. 
8 


295 


V9  a;  -  32 


4-  V  9  a;  -  32. 


8.  Va:  +  3  4-  Vx  +  S  -  V4  :c  +  21  =  0. 

9.  ^Vx+  0  -  ^Vx  —  5  =  y/2  Vi. 

Q 

10.  'V^m^  -f  a;  Vn^  +  x^  =  Vx-{-  vi;   Vx  +  Vx—2  =  —r' 

\  X 

11.  A^4+2V2^-5  =   V3;      V|  +  :^^m_ 

V  a:  —  V  m       ^ 


V2  ic  +  1  +  3  Va;  _        Vm  a;  —  n  __  3  v  m  a;  —  2  7i 
a/2  a;  +  1  —  3  Va;  Vwi x-\-  n      3  Vw  x  +  5n 


13 


V5a;+  VB       Va;4-  5      V6a;+2  _  4:V6x  +  6 
V3^  4-  V3  ~  V^  +  3'    V6^-  2  ~  4  V6'^ -  9 

14.  v/^+v/^^=c/4^- 

"w  +  a;       'm  —  a;       »7?r  —  ar 
Solve  the  following  for  x  and  y : 

15.  a;  +  4  V3  +  y  =  15  —  a;  4-  y  V5. 

16.  a;  4-  y  4-  a;  Va  4-  y  V^  =  1  —  Va. 

17.  a:  -  5  +  (2  y  -  3)  V3  =  5  a;  -  Vl2. 

18.  a;  —  rt  4-  {y  —  3) Va  4-  2>  =  w  a;  4-  Vol. 


19.  a  —  Va;  +  y  =  y  —  x  —  Vm  4-  71. 

20.  a;  Vw(Vm4- 1)  =  w  — wi4- y  V^(l  — V^. 


296  ELEMENTS  OF  ALGEBRA. 


CHAPTEK  XX. 
LOGARITHMS. 

119.  If  a^  =  m,  then  /  is  called  the  logarithm  of  m  to  the 
base  a.     Hence, 

A  Logarithm  is  the  exponent  by  which  a  certain  num- 
ber, called  the  base,  must  be  affected  in  order  to  produce  a 
given  number. 

The  logarithm  of  m  to  the  base  a  is  written  logam.  Thus, 
log„m  =  I  expresses  the  relation  a^  =  m;  logj^  100  =  2  expresses  the 
relation  10^  =  100,  etc. 

Since  numbers  are  formed  by  combinations  of  tens,  any  number 
may  be  expressed,  exactly  or  approximately,  as  a  power  of  10.     Thus, 


1000  =  103  .  ei-c 

120.  Common  System  of  Logarithms.  This  system  has 
10  for  its  base,  and  is  the  only  one  used  for  practical 
calculations.     Thus, 

Since  100=1,  log  1  =  0;  since  10^  =  10,  log  10  =  1  ; 

since      102  =  iqO,  log  100  =  2  ;  since  lO^  =  1000,  log  1000  =:  3; 

since      10*  =  10000,  log  10000  =  4  ;  and  so  on. 

Since  lO-i  =  J^  =  .1,  log  .1  =  -  1  =  9  -  10 ; 
since  10- 2  =  ^i^  =  .01,  log  .01  =  -  2  =  8  -  10  ; 
since       lO-s  =  ^Jq^  =  .001,  log  .001  =  -  3  =  7  -  10 ;  and  so  on. 

It  is  evident  that  the  logarithm  of  all  numbers  greater  than  1  is 
positive  J  and  of  all  numbers  between  0  and  1  is  negative ;  also,  that 
the  logarithm  of  any  numbers  between 


LOGARITHMS.  297 

1  and      W  is      0  -f  a  fraction; 

10  and    100  is      1  +  a  fraction  ; 
100  aiid  1000  is      2  +  a  fraction  ; 

1  and  .1  is  —  1  -f  a  fraction,  or  9  +  a  fraction  — 10; 
.1  and  .01  is  —  2  +  a  fraction,  or  8  +  a  fraction  — 10; 
.01  and  .001  is  —  3  +  a  fraction,  or  7  +  a  fraction  — 10;  and  so  on. 

It  thus  appears  that  the  logarithm  of  a  number  consists 
of  an  integral  part,  called  the  characteristic,  and  a  I'ractional 
part,  called  the  mantissa. 

The  mantissa  is  always  made  positive. 

IUu8tration8.  It  is  known  that  log  5  =  0.69897;  log  12  =  1.07918; 
log  2912  =  3.4(3419;    etc.      These  results  mean  that   loo«»8»'  =  5; 

1O1.07918  ^  12;    108.4M19  =  £912  ;    CtC. 

Notes :  1 .  'Hie  fractional  part  of  a  logarithm  cannot  be  expressed  exactly, 
but  an  apiiroxiniate  value  may  be  found,  true  to  as  many  decimal  places  as 
desire^l.  Thus,  the  logarithm  of  3  is  found  to  be  0.477121,  true  to  the  sixth 
place. 

2.  For  brevity  the  expression  "logarithm  of  3"  is  written  log 3.  The 
expression  "log  «"  is  read  "logarithm  of  x." 

3.  Logarithms  were  inventetl  by  John  Napier,  Baron  of  Merchiston,  Scot- 
land, and  first  published  in  1614. 

4.  Ih-re  are  only  two  systems  of  logarithms  in  general  use  :  the  Natural, 
or  Hyperbolic,  system,  and  the  lirhjtjsian,  or  Common,  system.  The  base  sub- 
script of  the  former  is  e,  and  that  of  the  latter  is  10. 

5.  The  natunil  system,  invente<l  by  Jolm  Speidell  and  published  in  1619,  is 
employetl  in  the  higher  branches  of  analysis  and  in  scientific  investigations  ; 
its  base  is  2.718281828+ . 

6.  The  common  system,  more  properly  called  the  denary  or  decimal  sys- 
tem, was  invente<i  by  Henry  Brigps,  an  Engli.sh  geometrician,  and  first  pub- 
lished in  1617.    The  logarithm  of  its  base,  10,  is  alrmys  1. 

7.  The  logarithms  invente<l  by  Napier  are  entirely  different  from  those  in- 
ventetl  by  Si>eidell,  though  they  are  closely  connected  with  them.  The  natural 
system  may  be  regardeil  as  a  modification  of  the  original  Napierian  system. 

121.  Since  log  1  =  0,  log  10  =  1,  log  100  =  2,  log  1000  =  3,  etc., 
the  characteristic  of  the  logarithms  of  all  numbers  consisting  of  one 


298  ELEMENTS  OF   ALGEBRA. 

integral  digit  (that  is,  all  numbers  with  one  figure  to  the  left  of  its 
decimal  point)  is  0;  of  all  numbers  consisting  of  two  integral  digits 
is  1 ;  of  all  numbers  consisting  of  three  integral  digits  is  2 ;  and  so 
on.     Hence, 

I.  The  characteristic  of  the  logarithm  of  an  integral 
number,  or  of  a  mixed  decimal,  is  one  less  than  the  numher 
of  integral  places. 

Since  log  .1  =  -  1,  log  .01  =  -  2,  log  .001  =  -  3,  etc. ;  the  charac- 
teristic of  the  logarithm  of  any  decimal  whose  first  significant  figure 
occupies  the  first  decimal  place  (that  is,  of  any  number  between  0.1 
and  1)  is  —  1 ;  of  any  decimal  whose  first  significant  figure  occupies 
the  second  decimal  place  (that  is,  of  any  number  between  0.01  and 
0.1)  is  —  2  ;  of  any  decimal  Vfho^Q  first  significant  figure  occupies  the 
third  decimal  place  (that  is,  of  any  number  between  0.001  and  0.01) 
is  —  3;  and  so  on.     Hence, 

II.  The  characteristic  of  the  loga7nthm  of  a  decimal  is 
negative,  and  is  numerically  equal  to  the  numher  of  the  place 
occupied  hy  the  first  aignificant  figure  of  the  decimal. 

The  characteristic  only  is  negative.  Hence,  in  the  case  of  decimals 
whose  logarithms  are  negative,  the  logarithm  is  made  to  consist  of  a 
negative  characteristic  and  a  positive  mantissa.  To  indicate  this,  the 
minus  sign  is  written  over  the  characteristic,  or  else  10  is  added  to 
the  characteristic  and  the  subtraction  of  10  from  the  logarithm  is 
indicated. 

Thus,  log  .0012  =  3.0792,  or  7.0792  -  10  ;  read  "characteristic 
minus  three,  mantissa  nought  seven  ninety-two,"  or  "characteristic 
seven  minus  ten,  etc."  In  reading  the  mantissa,  for  brevity,  two  inte»- 
gers  are  read  at  a  time.  Thus,  log  2  =  0.30103,  is  read  "the  loga- 
rithm of  two  equals  characteristic  zero,  mantissa  thirty  ten  three." 

Illustrations.  The  characteristic  of  the  logarithm  of  9  is  0; 
of  32  is  1 ;  of  433  is  2  ;  of  39562  is  4;  of  632.526  is  2 ;  of  .42  is  - 1  ; 
of  .023622  is  -  2 ;  of  .0000325  is  -  5  ;  etc. 


LOGARITHMS.  299 

122.  Let  m  and  n  be  any  two  numbers  whose  logarithms  are  x 
and  y  in  the  common  system.  Then  l(F  =  in  and  10*=  n.  Multiply- 
ing the  equations  together,  we  have  1(F+*'  =  mn.  Hence  (Art.  1 19), 
log  mn  =  z  +  y.  But  x  =  log  m  and  y  =  log  n.  Therefore,  log  m  n 
=  log  m  -H  log  n.  Similarly  log  in  np  =  log  m  -f  log  n  +  log  p  ;  etc. 
Hence, 

Tlie  logarithm  of  a  product  is  found  by  adding  together 
Die  logarithms  of  its  factors. 

Ulustrationa.  Given  l(»g  2  =  0.3010,  log  3  =  0.4771,  log  5 
=  0.6990,  log  7  =  0.8451. 

log  252  =  log  (2  X2X3X3X7) 

=  log  2  +  log  2  +  log  3  -f  log  3  +  log  7 

=  2  log  2  +  2  log  3  +  log  7 

=  2  X  0.3010  +  2  X  0.4771  +  0.8451 

=  0.6020  +  0.9542  +  0.8451 

=  2.4013. 

log  300  =  log  (2  X  3  X  5  X  10) 

=  log  2  +  log  3  +  log  5  +  log  10 
=  0.3010  +  0.4771  +  0.6990  +  1 
t=  2.4771. 

Exercise  106. 

Given  log  2  =  0.3010,  log  3  =  0.4771,  log  5  =  0.6990, 
log  7  =  0.8451 ;  find  the  values  of  the  following : 

1.  log  6;  log  64;  log  14;  log  8;  log  12;  log  15;  log  84. 

2.  log  343;  log  16;  log  216;  log  27;  log  45;  log  36. 

3.  log  90;  log  210;  log  3600;  log  1120;  log  1680. 

123.  If  any  number  be  multiplied  or  divided  by  any  integral 
power  of  10,  since  the  sequence  of  the  digits  in  the  resulting  number 
remains  the  same^  the  mantis-sa)  of  their  logarithms  will  be  unaffected. 
Thus,  since  it  is  known  that  log  577.932  =  2.7619, 


300  ELEMENTS  OF  ALGEBRA. 

log  5779.32  =  log  (577.932  X  10)       =  log  577.932  +  log  10 

2.7619  +  1  =  3.7619. 

log  57793.2  =  log  (577.932  X  100)     =  log  577.932  +  log  100 

2.7619  +  2  =  4.7619. 

log  57.7932  =  log  (577.932  X  0.1)     =  log  577.932  +  log  0.1 

2.7619  +  (-  1)  rr  1.7619. 
log  5.77932  =  log  (577.932  X  0.01)    =  log  577.932  +  log  0.01 

2.7619+ (-2)  =  0.7619. 

log  .577932  =  log  (577.932  X  0.001)  =  log  577.932  +  log  0.001 

=         2.7619  +  (-3)  =1.7619. 
Etc.     Hence, 

The  mantissce  of  the  logarithms  of  numbers  having  the 
same  sequence  of  digits  are  the  same. 

Illustrations.  If  log  44.068  =  1.6441,  log  4.4068  =  0.6441, 
log  .44068  =  1.6441  or  9.6441  -  10,  log  .000044068  =  5.6441  or 
5.6441  -  10,  log  440.68  =  2.6441,  log  440^800  =  6.6441,  etc.  If 
log  2  =  0.3010,  log  .2  =  1.3010,  log  .02  ^  2.3010,  log  20  =  1.3010, 
etc.     Hence, 

The  mantissa  depends  only  on  the  sequence  of  digits,  and 
the  characteristic  on  the  position  of  the  decimal  point. 


Exercise  107. 

1.  Write  the  characteristics  of  the  logarithms  of :  12753; 
13.2;  532;  .053;  .2;  .37;  .00578;  .000000735;   1.23041. 

2.  The  mantissa  of  log  6732  is  .8281,  write  the  logarithms 
of:  6.732;  673.2;  67.32;  .6732;  .006732;  .000006732.    . 

3.  Name  the  number  of  digits  in  the  integral  part  of  the 
numbers  whose  logarithms  are:  5.3010;  0.6990;  3.4771. 


LOGARITHMS.  301 

4.  Name  the  place  occupied  by  the  first  significant  fig- 
ure iu  the  numbers  whose  logarithms  are:  4.8451;  0.7782. 

Given  log  2  =  0.3010,  log  3  =  0.4771,  log  5  =  0.G990, 
log  7  =  0.8451;  find  the  logarithms  of  the  following 
number : 

5.  .18;  22.5;  1.05;  3.75;  10.5;  6.3;  .0125;  420. 

6.  .0056;   .128;    14.4;   1.25;   12.5;   .05;    .0000315. 

7.  .3024;   5.4;   .006;   .0021;   3.5;   .00035;  4.48. 

124.  Let  7«  be  any  number  whose  logarithm  is  x.  Then  UF^wi. 
Raising  both  members  to  the  pih  power,  we  have  10'"  =  inP.  Hence 
(Art.  119),  log  mf  =  px.  But  x  —  log?n.  Therefore,  log  Tn** 
=  -p  log  m.     Similarly  log  wi^'n'  =  p  log  m  -\-  q  log  n,  etc.     Hence, 

Tlie  logarithm  of  any  power  of  a  number  is  found  h/ 
multiplying  the  logarithm  of  tlie  number  by  the  exponent  of 
Vie  power. 

niustrations.     log  5"  =  10  log  5  =J0  X  0.6990  =  6.99(H), 
log  .003*  =  5  log  .(K)3  =  5  X  3.4771  =  13.3855. 
log  864  =  log  26  X  3«  =  5  log  2  +  3  log  3  =  5  X  0.3010  +  3X0  4771 
=  1.5060. 

Note.  If  the  number  i.s  a  decimal  and  the  exponent  positive,  the  j)roduct  of 
the  characteristic  and  exponent  will  be  negative,  and  since  the  mantissa  is  made 
positive,  we  must  algebraically  add  whatever  is  carried  from  the  niantis.sa. 

Thii.«»,  log  .0005"  =  10  X  4.6990  =  40  +  G.9900  =  34  +  0.9900  =  34.9900. 

Exercise  108. 

Given  log  2  =  0.3010,  log  3  =  0.4771,  log  5  =  0.6990, 
log  7  =  0.8451 ;  find  the  logarithms  of: 

1.  2*;   53;   7^;    8^   3^;   64;   81;   72;    (8.1)7;   (2.10)6 

2.  343;   .036;   .000128;  (.0336)1^   (.00174)2;   (3.84)» 


302  ELEMENTS  OF  ALGEBRA. 

125.  Let  m  and  n  be  any  two  numbers  whose  logarithms  are  x 
andy.     Then  10^  =  m  and  10^' =  w.     .-.   10^-2' =  m -f  n. 

m  "  mn 

log  -  =  a:  -  2/  =  log  m  -  log  n.     Similarly,  log  ^^^  =  log  m  +  log  n 

—  (log  wij  +  log  Uj).     Etc.     Hence, 

The  logarithm  of  a  quotient  is  found  by  subtracting  the 
logarithm  of  the  divisor  from  the  logarithm  of  the  dividend. 

Illustrations,     log  |  =  log  3  -log  2  =  0.4771-0.3010  =  0.1761. 
log  f  =  log  5  -  log  7  =  (0.6990)  -  0.8451  =  (1.6990  -  1)  -  0.8451 
=  0.8539  -  1  =  1.8539. 

Note.  To  subtract  a  greater  logarithm  from  a  less  logarithm.  Add  to  the 
characteristic  of  the  minuend  the  least  number  which  will  make  the  minuend 
greater  than  the  subtrahend  ;  also  indicate  the  subtraction  of  the  same  number 
from  the  minuend  so  increased.    Then  proceed  as  before.    Thus, 

log  =^  =  log  252  -  log  300  =  (2.4014)  -  2. 4771  =  (3.4014-1)  -  2. 4771  =  1.9243. 

log  '-^  =  (3.6990)  -  2T8451  =  (T.6990  -  2)  -  2.8451  =  0.8539-2  =  2.8539  or 
8.8539  - 10. 

Exercise  109. 

Given  log  2  =  0.3010,  log  3  =  0.4771,  log  5  =  0.6990, 
log  7  =  0.8451 ;  find  the  logaritlims  of: 


1    ^ 

2' 


_3^     7     3  .003     .005      /SV     .007 

'    .05'    5'    5'      2'      Q7  ;      02  '    VlO/   '     -02' 


2    42.     -:!!_.    125-    '^-    ^-     ^i^.    5.  ^ 
■     ^'    .0052'  '      5  '     8.1'    .000027'       '  .007* 

126.    Let  m  be  any  number  of  which  the  logarithm  is  x.     Then 

X  

1(F  =  m.     Taking  the  rth  root  of  each  member,  we  have  10'"  =  \/m. 
.'.    (Art.    119),   log    \/m  —  -  =   -^ — .      Similarly,  log    ^^m  n 


LOGARITHMS.  303 

The  logarithm  of  any  root  of  a  numher  is  found  hy  divid- 
ing the  logarithm  of  the  number  by  tJie  index  of  the  root. 

«/-      log  5      0.6990     ^   _„ 
IHuatrations.     log  ^  =  -|-  =  — 5—  =  0.1398. 

,, log.0(X)7      4.8451      3.8451  +  7  ^„   -     -,,^« 

log  '^/ioOO?  =  -^ =  -y—  = 7-^  =  0.5493+1  =  1.5493. 

1        VT-^.      logl5<^      5  log  3  X  .5       5  (log  3  +  log  .5) 
logVl-5»=— ^r— =  e = 6 

^5(0.4771  +  1.6990)^^^^^^^ 
o 

Note.  If  a  negative  characteristic  is  not  exactly  divisilile  by  tlie  index  of 
the  root,  subtract  from  the  characteristic  the  least  positive  number  which  will 
make  it  so  divisible.  Indicate  the  addition  of  the  characteristic  so  formed  to 
the  mantissa,  and  prefix  the  number  subtracted  from  the  characteristic  to  the 
mantissa.    Then  divide  separately.    Thus, 

log  V75  =  ?5|:5  =  I:^  =  '•6»y  +  ^  =  0.8495  +  T  =  T.8495  or  9.8495  -  10. 

Ik  ii  it 


Exercise  110. 

Given  log  2  =  0.3010,  log  3  =  0.4771,  log  5  =  0.6990, 
log  7  =  0.8451  ;  find  the  logarithms  of: 

1.  ^7;    \^l;   \/2-   v^.^;   ^243;  ^12^;  \^M;  ^^. 

2.  ^^iTK2f;  5ix3i;  ^,;  g;  ^^^;   6i  x  3f. 

,,     ^^x\/2      j^^    ^      7/  J.21)2_       V^2^ 
"•   -;/l8xV2^    ^15'   -^!^7'   V(.()()084)2'     ^^^^^fg ' 

127.  Table  of  Logarithms.  The  table  (pages  304  and  305) 
gives  the  nianti.Hsae  of  the  logarithms  to  four  decimal  i>]jices  for  all 
numbers  from  1  to  10(X)  inclusire.  The  characteristic  and  decimal 
points  are  omiUedy  and  must  be  supplied  by  inspection  (Art.  121. 


304 


ELEMENTS  OF  ALGEBRA. 


N 

0    1 

2 

3 

4    6 

6 

7 

8 

9 

10 

11 
12 
13 
14 

ouoo 

0414 
0792 
1139 
1461 

0043 
0453 
0828 
1173 
1492 

0086 
0492 
0864 
1206 
1523 

0128 
0581 
0899 
1239 
1558 

0170 
0569 
0934 
1271 
1584 

0212 
0607 
0969 
1803 
1614 

0253 
0645 
1004 
1335 
1644 

0294 
0682 
1038 
1367 
1673 

0384 
0719 
1072 
1899 
1703 

0874 
0755 
1106 
1480 
1732 

15 

16 
17 
18 
19 

1761 
2011 
2304 
2553 

2788 

1790 
2068 
2330 

2577 
2810 

1818 
2095 
2355 
2(501 
2833 

1847 
2122 
2380 
2625 

2856 

1875 
2148 
2405 
2648 

2878 

1908 
2175 
2430 

2672 
2900 

1931 
2201 
2455 

2695 
2928 

1959 

2227 
2480 
2718 
2945 

1987 
2253 
2504 
2742 

2967 

2014 
2279 
2529 
2765 
2989 

20 

21 
22 
23 
24 

3010 
8222 
8424 
3617 

3802 

;;o32 

3243 
3444 
3636 
3820 

3054 
8263 
8464 
3655 

3838 

3075 
3284 
3488 
3674 
3856 

3096 
3304 
8502 
3692 

3874 

3118 
3324 
3522 
8711 
3892 

3139 
3845 
8541 
3729 
3909 

3160 
3365 
3560 
3747 
3927 

3181 
3385 
3579 
3766 
3945 

8201 
3404 

3598 
3784 
3962 

25 

26 
27 
28 
29 

3979 
4150 
4814 
4472 
4624 

8997 
4166 
4330 

4487 
4639 

4014 
4183 
4346 
4502 
4654 

4031 
4200 
4362 
4518 
4669 

4048 
4216 
4378 
4533 
4683 

4065 
4282 
4393 
4548 
4698 

4082 
4249 
4409 
4564 
4713 

4857 
4997 
5132 
5263 
5891 

4099 
4265 
4425 
4579 

4728 

4116 
4281 
4440 
4594 
4742 

4133 
4298 
4456 
4609 
4757 

30 

31 
32 
33 
34 

4771 
4914 
5051 
5185 
5315 

4786 
4928 
5065 
5198 
5328 

4800 
4942 
5079 
5211 
5340 

4814 
4955 
5092 
5224 
5353 

4829 
4969 
5105 
5237 
5866 

4843 
4983 
5119 
5250 

5378 

4871 
5011 
5145 
5276 
5403 

4886 
5024 
5159 
5289 
5416 

4900 
5038 
5172 
5302 
5428 

35 

36 
37 
38 
39 

5441 
5568 

5682 
£798 
5911 

5453 
5575 
5694 
5809 
5922 

5465 
5587 
5705 
5821 
5933 

5478 
5599 
5717 
5882 
5944 

5490 
5611 
5729 
5848 
5955 

5502 
5623 
5740 
5855 
5966 

5514 

5635 
5752 
5866 
5977 

5527 
5647 
5763 

5877 
5988 

5539 
5658 
5775 
5888 
5999 

5551 
5670 
5786 
5899 
6010 

40 

41 
42 
43 
44 

6021 
6128 
6282 
6335 
6435 

6031 
0138 
6243 
6345 
6444 

6042 
6149 
6253 
6355 
6454 

6053 
6160 
6263 
6365 
6464 

6064 
6170 
6274 
6375 
6474 

6075 
6180 
6284 
6385 
6484 

6085 
6191 
6294 
6395 
6493 

6096 
6201 
6304 
6405 
6508 

6107 
6212 
6314 
6415 
6513 

6117 
6222 
6825 
6425 
6522 

45 

46 
47 
48 
49 

6532 
6628 
6721 
6812 
6902 

6542 
6637 
6730 
6821 
6911 

6551 
6646 
6789 
6830 
6920 

6561 
6656 
6749 
6889 
6928 

6571 
6665 
6768 
6848 
6937 

6580 
6675 
6767 
6857 
6946 

6590 
6684 
6776 
6866 
6955 

6599 
6693 
6785 
6875 
6964 

6609 
6702 
6794 
6884 
6972 

6618 
6712 
6803 
6893 
6981 

50 

51 
52 
53 
64 

6990 
7076 
7160 
7243 
7324 

6998 
7081 
7168 
7251 
7382 

7007 
7093 
7177 
7259 
7340 

7016 
7101 

7185 
7267 
7348 

7024 
7110 
7198 
7275 
7356 

7033 
7118 
7202 
7284 
7364 

7042 
7126 
7210 
7292 
7372 

7050 
7135 
7218 
7300 
7380 

7059 
7148 

7226 
7308 
7388 

7067 
7152 
7285 
7316 
7396 

LOGARITHMS. 


305 


N 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

55 

56 
67 
68 
69 

7404 
7482 
7559 
lOU 
7709 

7412 
7490 
7566 
7642 
7716 

7419 
7497 
7574 
7(W9 
7723 

7427 
7505 
7582 
7657 
7731 

7435 
7513 
7689 
7664 
7738 

7448 
7520 
7597 
7672 
7745 

7451 
7528 
7004 
7079 
7752 

7459 
7636 
7612 
7686 
7760 

7466 
7643 
7610 
7694 
7767 

7474 
7661 
7627 
7701 

7774 

00 

61 
62 
63 
64 

7782 
7853 
7924 
7993 
8062 

7789 
7800 
7931 
8000 
8069 

7796 
7868 
7938 
8007 
8075 

7803 
7875 
7945 
8014 
8082 

7810 
7882 
7952 
8021 
8089 

7818 
7889 
7959 
8028 
8096 

7825 
7896 
7966 
8035 
8102 

7832 
7903 
7973 
8041 
8109 

7889 
7910 
7980 
8048 
8116 

7846 
7917 
7987 
8056 
8122 

65 

66 
67 
68 
69 

8129 
8195 
8261 
8325 
8388 

8136 
8202 
8267 
8331 
8895 

8142 
8209 
8274 
8338 
8401 

8149 
8215 
8280 
8344 
8407 

8156 
8222 
8287 
8351 
8414 

8162 
8228 
8293 
8357 
8420 

8169 
8236 
8299 
8363 
8426 

8176 
8241 
8306 
8370 
8432 

8182 
8248 
8812 
8376 
8430 

8189 
8264 
8819 
8382 
8446 

70 
71 
72 
78 

74 

8451 
8513 
8573 
8633 
8692 

8457 
8519 
8579 
8639 
8098 

8463 
8525 
8586 
8645 
8704 

8470 
8531 
8691 
8651 
8710 

8768 
8825 
8882 
8938 
8993 

8476 
8637 
8697 
8657 
8716 

8482 
8643 
8603 
8663 
8722 

8488 
8549 
8609 
8669 
8727 

8494 
8555 
8015 
8675 
8733 

8600 
8661 
8621 
8681 
8739 

8606 
8667 
8627 
8686 
8746 

75 

76 

77 
78 
79 

8761 

8808 
8866 
8921 
8976 

8756 
8814 
8871 
8927 
8982 

8702 
8820 
8876 
8932 
8987 

8774 
8831 
8887 
8943 
8998 

8779 
8837 
8893 
8949 
9004 

8786 
8842 
8899 
8954 
9009 

8791 
8848 
8904 
8960 
9015 

8797 
8864 
8910 
8966 
9020 

8802 
8869 
8916 
8971 
0025 

80 
81 
82 
83 
84 

9031 
9086 
9138 
9191 
9243 

9036 
9090 
9143 
9196 
9248 

9042 
9096 
9149 
9201 
9263 

9047 
9101 
9164 
9206 
9268 

9053 
9100 
9150 
9212 
9263 

9058 
9112 
9106 
9217 
9269 

9063 
9117 
9170 
9222 
9274 

9069 
9122 
9176 
9227 
9279 

9074 
9128 
9180 
9232 
9284 

9079 
9183 
0186 
0238 
9289 

85 
86 
87 
88 
89 

9294 
9346 
9396 
9446 
9494 

9299 
9360 
9400 
9460 
9499 

9304 
9355 
9405 
9456 
9604 

9309 
9360 
9410 
9460 
9600 

9316 
9365 
9416 
9465 
9613 

9320 
9370 
9420 
9469 
9618 

9325 
9375 
9426 
9474 
9628 

9330 
9380 
94:^ 
9479 
9528 

9836 
0386 
0435 
9484 
9533 

9840 
9390 
9440 
9489 
9638 

90 
91 
92 
93 
94 

9642 
9690 
9688 
9686 
9731 

9647 
9696 
9643 
9689 
9786 

9652 
96C0 
9647 
9694 
9741 

9667 
9606 
9662 
9699 
9746 

9662 
9609 
9667 
9703 
9760 

9666 
9614 
9661 
9708 
9764 

9571 
9019 
9666 
9713 
9759 

9576 
9624 
9671 
9717 
9763 

9581 
9628 
0675 
9722 
9708 

9686 
9633 
9680 
9727 
9773 

95 

96 
97 
98 
99 

9777 
9823 
9868 
9912 
9966 

9782 
9827 
9872 
9917 
9961 

9786 
9832 
9877 
9921 
9966 

9791 
98:^6 
9881 
9926 
9969 

9796 
9841 
0886 
0930 
9074 

0800 
9846 
0890 
9984 
9078 

9805 
9850 
9894 

9083 

0809 
9864 
0809 
0943 
0987 

0814 
9859 
91K)3 
9948 
99*^1 

9818 
0863 
9908 
0962 
0906 

20 


a06  ELEMENTS   OF  ALGEBRA. 

Explanation  of  Table.  The  left-hand  column,  headed  N,  is  a 
column  of  numbers.     The  figures  O,  1,  2,  3,  4,  5,  6,  7>  8,  9, 

opposite  N  at  the  top  of  the  table,  are  the  right-hand  figures  of  num- 
bers whose  left-hand  figures  are  given  in  the  column  headed  N.  The 
figures  in  the  column  which  they  head  are  the  corresponding  man- 
tissse  of  the  logarithms  of  the  numbers. 

128.   To  Find  the  Logaritlmi  of  a  Number. 

I.  Consisting  of  one  Figure.  The  mantissae  of  the  logarithms 
of  single  digits,  1,  2,  3,  4,  etc.,  are  seen  opposite  10,  20,  30,  40,  etc., 
and  in  the  column  headed  O.  To  the  mantissa  prefix  the  character- 
istic and  insert  the  decimal  point.     Thus, 

log  6  =  0.7782.    log  .6  =  1.7782.    log  8  =  0.9031. 

Similarly,  since  the  mantissa  of  log  .009  is  the  same  as  the  man- 
tissa of  log  9,  log  .009  -  3.9542. 

II.  Consisting  of  t-wo  Figures.  In  the  column  headed  N  look 
for  the  figures.  In  the  line  with  the  figures,  and  in  the  column 
headed  0,  is  seen  the  mantissa.     Then  proceed  as  before.     Thus, 

log  13  =1.1139.     log  2.5  =  0.3979.     log  .92  =  7.9638. 

Similarly,  log  .00092  =  4.9638. 

III.  Consisting  of  three  Figures.  In  the  column  headed  N, 
look  for  the  first  two  figures,  and  at  the  top  of  the  table  for  the  third 
figure.  In  the  line  with  the  first  two  figures,  and  in  the  column 
headed  by  the  third  figure,  is  seen  the  mantissa.  Then  proceed  as 
before.     Thus, 

•  log  313  =  2.4955.     log  17.9  =  1.2529.     log  .279  =  T.4456. 
Similarly,  log  .000718  =  4.8561. 

IV.  Consisting  of  more  than  three  Figures,  Take  the  man- 
tissa of  the  logarithm  of  the  first  three  figures  as  given  in  the  table. 
Prefix  a  decimal  point  to  the  remaining  figures  of  the  number,  and 
multiply  the  result  by  the  tabular  *  difference.     Add  the  product  to 

*  The  Tabular  difference  is  the  difference  between  the  two  successive  raan- 
tissjfi  between  which  the  required,  or  given,  mantissa  Ues. 


LOGARITHMS.  307 

the  mantissa  thus  taken.     Prefix  the  characteristic  and  insert  the 
decimal  point  as  before.     Thus, 

1.   Find  the  logarithm  of  80.672. 

The  tabular  mantissa  of  the  logarithm  of  806  is  9063 

The  tabular  mantissa  of  the  logarithm  of  807  is  9069 

Therefore,  the  tabular  difference  =  6 

The  number  80672  being  between  80600  and  80700,  the  mantissa 
of  its  logarithm  must  be  between  9063  and  9069.  An  increase  of  100 
in  80600  causes  an  increase  of  6  in  the  mantissa  of  the  logarithm  of 
80600.  Therefore,  an  increase  of  72  in  80600  will  produce  an  increase 
of  ^  of  6  (or  .72  X  6),  or  4.32,  in  the  mantissa  of  the  logarithm  of 
80600.  Hence,  the  tabular  mantissa  of  log  80672  mtist  be  9063  +  4, 
or  9067.  Prefixing  the  characteristic  and  inserting  the  decimal  point, 
we  have 

log  80.672  =  1.9067. 

Similarly,  since  the  mantissa  of  log  .0005102  is  the  same  as  the 
mantissa  of  log  5102, 

2    To  find  the  logarithm  of  .0005102. 
The  tabular  mantissa  of  log  510  is  7076 

The  tabular  mantissa  of  log  511  is  7084 

.*.  the  tabular  diflference  =  8 

Hence,  the  tabular  mantissa  of  log  6102  must  be  7076  -f  .2  X  8,  or 
7078. 

.-.  log  .0005102  =  4.7078. 

Exercise   111. 

Find  by  means  of  the  table  the  logarithms  of  the 
following : 

1.  70;    102;    201;    999;    .712;    3.6;    .00789;    3.21. 

2.  .0031;    .0983;    .00003;    10.08;    29461;    3015.6. 

3.  32678;    V337;    ^/Msm2;    4^| ;    (.098x85)*. 


308  ELEMENTS  OF  ALGEBRA. 

129.      To  Find  a  Number  when  its  Logarithm  is  Given. 

I.  If  the  Given  Mantissa  is  Found  in  the  Table.  The  first 
two  figures  of  the  required  number  will  be  seen  on  the  same  line 
with  the  mantissa  and  in  the  column  headed  N,  the  third  figure  will 
be  seen  at  the  head  of  the  column  in  which  the  mantissa  is  found. 
Finally  insert  the  decimal  point  as  the  characteristic  directs.     Thus, 

1.  Find  the  number  whose  logarithm  is  1.9232. 

Look  for  9232  in  tbe  table.  It  is  found  on  the  line  with  83  and 
in  the  column  headed  8.  Therefore,  write  838  and  insert  the  deci- 
mal point.     Hence,  the  number  required  is  .838. 

II.    If  the  Given  Mantissa  cannot  be  Found  in  the  Table. 

Find  the  next  less  mantissa,  and  the  corresponding  number ;  also  find 
the  tabular  diff'erence.  Annex  the  quotient  of  the  difference  between 
the  given  mantissa  and  the  next  less  mantissa  divided  by  the  tabular 
difference,  to  the  corresponding  number ;  then  proceed  as  before. 
Thus, 

2.  Find  the  number  whose  logarithm  is  2.7439. 

The  next  less   mantissa  is  7435,  corresponding  to  554. 

The  next  greater  mantissa  is  7443,  corresponding  to  555. 

.-.  the  tabular  difference  =  8. 

The  diflference  between  the  given  mantissa  and  the  next  less  man- 
tissa is  4.  Since  the  given  mantissa  lies  between  7435  and  7443,  the 
corresponding  number  must  lie  between  554  and  555.  An  increase  of 
8  in  the  mantissa  causes  an  increase  of  1  in  554.  Therefore,  an  in- 
crease of  4  in  the  mantissa  will  produce  an  increase  of  ^,  or  .5,  in  554. 
Hence,  the  mantissa  7439  must  correspond  to  the  number  554+  .5,  or 
554.5.  Therefore  (II,  Art.  121),  write  05545  and  prefix  the  decimal 
point.     Hence,  the  number  required  is  .05545. 

3.  Find  the  number  whose  logarithm  is  3.1658. 

The  next  less  mantissa  is  1644,  corresponding  to  146. 

The  next  greater  mantissa  is  1673,  corresponding  to  147. 

.•.  the  tabular  difference  =         29. 

The  difference  between  the  given  mantissa  and  the  next  less  man- 
tissa is  14.  Annex  \^,  or  .48  nearly,  to  the  number  146,  and  insert 
the  decimal  point  as  the  characteristic  directs.  Hence,  the  number 
required  is  1464,8. 


LOGARITHMS.  309 

Exercise  112. 
Find  the  numbers  whose  logarithms  are : 

1.  3.4683;    2.4609;    4.8055;    0.4984;    0.1959. 

2.  3.6580;    2.4906;    4.5203;    2.5228;    0.6595. 

3.  0.8800;    1.7038;    5.8017;    3.1144;    5.7319. 

130.  An  Exponential  Equation  is  one  in  which  the  expo- 
nent is  the  unknown  number  ;  as,  iif  =  n,  ifrf  =  n.  Such 
equations  usually  require  logarithms  for  their  solutions. 

Example  1.    Solve  the  equation  2F  =  1.5. 

Process.   Take  the  logarithm  of  each  member,  x  log  21  =  log  1 .6. 

By  means  of  the  table,  1.3222  x  =  .1761. 

.1761 
Therefore,  x  =  .  ^^aa  =  .1332,  nearly. 

Example  2.     Find  the  value  of  3.208  X  .0362  X  .15734. 

Process,  log  (3.208  X  .0362  X  .15734)  =  log  3.208  +  log  .0362 
+  log  .15734. 

log  3.208  =0.5062 
log  .0362  =2.5587 
log  .16734  =  1.1969 

2.2618  =  log  .01827. 
Therefore,  3.208  X  .0362  X  .15734  =  .01827. 

Example  3.    Find  the  fifth  root  of  .05678. 
Process,     log  .05678  =  2.7542. 
5)2.7542  =  5)3.7542  +  5 

.7508  +  T  =  T.7508  =  log  .6634,  nearly. 


Therefore,  -y/.05678  =  .5634,  nearly. 


310  ELEMENTS  OF  ALGEBRA. 

Example  4.    Find  the  value  of  log^^  144. 

Solution.    To  find  loggi/g  144,  is  the  same  as  solving  (Art.  119) 
(21/3)'  =  144,  for  I,  squaring  each  side,  etc.,  I  =  4. 
Therefore,  logg  ^3 144  =  4. 

Exercise  113. 

Find  by  logarithms  the  values  of  the  following  : 

1.    360  X. 0827;    117.57  X  .0404  ;    i^  ;    (31.89)3 


2.   ^951;  380.57  X  .000967;  ^(•"^^^);^.y-Q"°^^^^ 

212.6  X  30.2         7435      -^343 
84.3  X  3.62  X  .05632'  38731  X  .3962'  ^f2^' 


4.  —  ■  •    ^ ;  72132  X  .038209  ;  -7.000313. 

^385.67 

5.  (61173)*;  -^;    ^;  ^X^.00l;   Ip. 
^  ^  '  (.19268)i      v'27  5^49 

Solve  the  followiug  equations  : 

6.  20"  =  100  ;    2"  =  769 ;    10"  =  4.4  ;    {^Y  =  17.4. 

7.  10^  =  2.45 ;    5'-'"  =  2"+^ ;  a/S^^^  =  '^/W^. 

8.  2*  X  6*-2  =  52*  X  7'-" ;    3^-'  =  5  ;    4"  =  64. 

9.  (1)"  =10;    rrf  =  n\    m"*+*  =  71 ;  ?/i""'  X  c^"'  =  n. 

10.  2^+^  =  6^  3^  =  3x2^+^  31-0^-1^  =  4-1.^  2^*-^  =  3^^-''. 

11.  a2*^3y  =  ^5  ^3x^2.  :^  ^^0.  ^x,^5y  =  (^7)4  ^^^  =  (^y)8 


LOGARITHMS.  311 

Find  the  number  of  digits  in  the  values  of: 

12.  312x28;    2^4;    W^o .    (4375)8.    (396000)io. 

Find  the  number  of  ciphers  between  the  decimal  point 
and  the  first  significant  figure  in  the  values  of: 

13.  (.2)*;    (.5y«>;    (.05)5;    (.0336)io ;    "x/sm, 

14.  Given  log  x  =  2.30103,  find  log  xi 

Find  the  values  of : 

15.  loga  4  ;    loga  8  ;    log^  32  ;    log^  128  ;    log,  1024. 

16.  log2  J  ;    log2  J  ;    logj  ^  ;    log^  ^^  ;    log^  \^16. 

17.  logs  729;   log5l25;   log,  625  ;   log,  15625;    log,  J. 

18.  log_e  1296  ;    log_.  -  ^{^  ;    log.  ^{-^ ;    logg^  512. 

19.  logs ^5- 125;    log848  49;    log8l28;    loga^/s  tJi- 

20.  log,|/a^^^;    log27^\;    logj4;    log,  a ;    log„|. 

21.  If  8  is  tlie  base,  of  what  number  is  §  the  logarithm? 
Of  what  J  ?  Of  what  1  ?  Of  wliat  2  ?  Of  what  3  ?  Of 
what  If  ?     Of  what  2  J  ?     Of  what  3  J  ?     Of  what  Y  ^^  ? 

22.  In  the  systems  whose  bases  are  10,  3,  and  J,  of  what 
numbers  is  —  5  the  logarithm  ? 

Find  the  bases  of  the  systems  in  which  : 

23.  log  81  =  4;  log  81  =  -  4 ;  log  j^^  =  4; 
log  iiftW  = -4;  log  i^  =  ±  2;  log  1024- =  ±571. 


312  ELEMENTS  OF  ALGEBRA. 


CHAPTER  XXI. 


QUADRATIC   EQUATIONS, 

131.  A  Quadratic  Equation  is  an  equation  in  which  the 
square  is  the  highest  power  of  the  unknown  number. 

A. Pure  Quadratic  Equation  is  an  equation  which  contains 
only  the  square  of  the  unknown  number;  as,  5  x^  =  17. 

An  Affected  Quadratic  Equation  is  an  equation  which 
contains  both  the  square  and  the  first  power  of  the  un- 
known number ;  as,  5  ^^  —  2  a^  =  10. 

o  ,      ^"^  +  5      ^        a;      17 
Example.    Solve  —  I  ^  =  o  +  "t-  • 

Process.     Clearing  of  fractions,  12  a;^  +  60  —  9  a;^  =  4  a:^  -f  51. 
Transposing  and  uniting,  x-  =  9. 

Therefore,  extracting  the  square  root,*  x  =  ±  S.     Hence, 

To  Solve  a  Pure  Quadratic  Equation.     Find  the  value  of  the 

square  of  the  unknown  number  by  the  method  for  solving  a  simple 
equation,  and  then  extract  the  square  root  of  both  members. 

Note.  *  In  extracting  the  square  root  of  both  members  of  the  equation 
a;2  z=  9,  we  ought  to  prefix  the  double  sign  (±)  to  the  square  root  of  each  mem- 
ber; but  there  are  no  new  results  by  it,  and  it  is  sufficient  to  w)1te  the  double 
sign  before  one  member  only.  Thus,  if  we  write  ±,  x  =  ±  S,  we  have  +  x 
=  4-3,  -^  X  =  —  S,  — aj  =  -j-3,  and  —  x  =  —  3;  but  the  last  two  become 
identical  to  the  first  two  on  changing  the  signs  of  both  members.  So  that  in 
either  case,  x  =  3,  and  a?  rr  —  3. 


QUADRATIC  EQUATIONS.  313 

Exercise  114. 
Solve  the  foUowiug  equations : 

o 

3.   a;(a;-10)  =  (6|-a:)10;    {6 x  +  ^f  =  756^  +  5x. 

35  -  2  a;      5x^  +  7  __  17_-Jj; 
9         "*"5a,-2-7~        3 

7.  I  (2  a;  -  5)2  =  94  -  24x;    3  ar^  -  4  =  ^±j?  . 

8.  -o =  -o >    rt  ar  +  7yj  it'  =  a  c^  +  ??i  x. 

3r  —  n      or  —  m 

9.  2:2  ^  ^^aj_^  =  ^j2:(l  —  wa;);    2  +  4.i'2  _.  ^^(1  __^ 


10.  7^  —  n  X  ■\-  m  =  n X (x  —  \)',    x  VB  +  x^  =  1  +  x^. 

--     mn  —  ic      ?i  —  aa;  .      .-^ 5 

11.    = ;    X  +  vx^  —  .3  = 


n  —  mx      an  —  x  ^x^  —  3 

12       ^      I      ^     ^,.  1  ,  1 ^^^ 

•  5+^      5_a;        >  1_>/1_^       l+Vl-a;2        a:2 


314  ELEMENTS  OF  ALGEBRA. 

132.     Example  1.     Solve  x^  —  2ax  +  4ab=:  2bx. 

Solution.  Transposing,  we  have  x^  —  2ax  +  4ab  —  2bx=  0. 
Arrange  in  binomial  terms  and  factor,  and  we  have  (x  —  2  a)  (x  — 26) 
=  0.  A  product  cannot  be  zero  unless  one  of  the  factors  is  zero. 
Hence,  the  equation  is  satisfied  if  x  —  2  a  =  0,  or  x  —  2  6  =  0  ;  that 
is,  a  x  =  2ay  or  if  X  =  2  b. 

Example  2.     Solve  ^  x2  +  f  x  -f-  20^  =  42f  +  x. 

Process.     Clear  of  fractions,  transpose,  and  unite, 
3  x2  -  2  X  -  133  =  0. 
Factor,  (3  x  +  19)  (x  -  7)  =  0. 

Therefore,  3  x  +  19  =  0,  and  x  -  7  =  0.  x  =  -  6^,  and  x  =:  7. 
Hence, 

To  Solve  a  Quadratic  Equation  by  Factoring.  Simplify  the 
equation,  with  all  its  terms  in  the  first  member  ;  then  place  the  fac- 
tors of  the  first  member  separately  equal  to  zero,  and  solve  the  simple 
equations  thus  formed. 

Exercise  115. 

Solve  the  following  equations  : 

1.  rz;2_i0;i;=:24;   a?2+2a;  =  80;    ^2_  iga?  +  32  =  0. 

2.  a;2  +  10   =  13  (x  +  6) ;  a^  +  4x  -  50  =  2  -  5  X, 

3.  4x2+13^+3  =  0;    Zx^ -^  1  = -\\x  -  x^  ^  ^. 

4.  a;2-ic=  11342;    5a^+3x-4==8aj-7rz;2_2. 

5.  \-Zx-x^  =  2x--^x-Z\    x^-2ax-\-%x  =  \^a. 

6.  :i;3_5^2^5^  + 7^2.    a^_|^  +  _9_^0. 
x-\-Z      2x-Z      3-x 


7.  lla:2_iii  =9^. 


x+  2        x-1        x-^2 


QUADRATIC  EQUATIONS.  315 

8.    (a;-2)(a;H9a:  +  20)  =  0;    2x^  +  Sa^^2z-S  =  0. 

3ar— 4  3a  —  2a;       4 


10.    mqx^  —  mnx-\-pqx  —  n2y  =  0;   x+5  =  Vx+ 5-\-6. 


11.    {a-b)x^-(a  +  b)x  +  2b  =  0.;   x^=21  +  Vx^-^' 

133.  An  aflfected  quadratic  equation  can  always  be  solved  by  the 
method  of  completing  the  square.  This  method  consists  in  adding  to 
both  memlxjrs  such  an  expression  as  will  make  the  memlier,  with  all 
the  terms  containing  the  unknown  number,  a  perfect  square.  The 
explanation  of  this  methocl  depends  upon  the  principle  that  a  trino- 
mial is  a  perfect  st^uare  when  one  of  its  terms  is  plus  or  minus  twice 
the  product  of  the  scjuare  roots  of  the  other  two.  This  process 
enables  us  to  extract  the  square  root  of  the  member  containing  the 
unknown  number,  and  thus  form  two  simple  equations  which  may 
be  solved  separately. 

2x-  11 
Example.    Solve  ^(S-x)  -    ^^    =  i  (a?  -  2). 

Process.     Clear  of  fractions,  transpose,  and  unite, 

-4x^  +  26x=  12. 
Divide  by -4,  ar»-J^a:  =  -3. 

Add  * (l^)« to lioth  members,  x^^^x  +  (J^f  =  -  3  +  (J^y  =  ^. 
Extract  the  square  root,  x  —  ^z=±^. 

Therefore,  x-^  =  ^,  &nd  x-^  = -^.     x  =  6,  and  x  =  f 

Every  affected  quadratic  equation  may  be  reduced  to  the 
general  form 

7Ji2^  +  7ix  -\-  a  =  0; 

where  m,  n,  and  a  represent  any  numbers  whatever,  positive  or 
negative,  integral  or  fractional.     Dividing  both  members  by  m  for 

a  n 

convenience,  representing  -  by  6  and  -  by  c,  and  transposmg,  we 

have 


316  ELEMENTS  OF  ALGEBRA. 

x^  -\-  c  X  =  —  b. 
Add    (2)    to  both  members,  x^  +  cx+i^j    =-^+(2)- 

Or,  x^+cx+  (0  =|(c2-4i). 

c  

Extract  the  square  root,  x  +  ^  =  ±^  a/c'^  —  4 6. 

Therefore,  x  +  ^  =  ^  ^.c^  -4  b,  and  x  +  ^  =  -^  ^c^-  4  6.    From 


which  a:  =  -  9  +  |  V^'"^  "  'I  ^'  ^"^  ^  =  ~  2  ~  ^  V^^  ~  ^  ^-     "^^^^^ 


-c-l-Vc2-46 
values  may  be  written  in  the  form  x  — 5 •     Hence, 


Common  Method  of  Solving  Quadratics.  Reduce  the  equa- 
tion to  the  form  x^  +  c  x  =  —  b.  Comflete  the  square  of 
the  first  member  by  adding  to  each  member  of  the  equation 
the  square  of  half  the  coefficient  of  x.  Extract  the  square 
root  of  both  member's,  and  solve  the  resulting  simple 
equations. 


Notes :  1.  *  Always  indicate  the  square  of  the  expression  to  be  added,  in  the 
first  member. 

2.  Since  the  squared  terms  of  the  square  of  a  binomial  are  ahvays  positive, 
the  coefficient  of  x"^  must  be  made  +  1,  if  necessary,  before  completing  the 
square.     This  may  be  done  by  multiplying  or  dividing  both  members  by  —  1. 

3.  The  foregoing  method  is  called  the  Italian  Method,  having  been  used 
by  Italian  mathematicians,  who  first  introduced  a  knowledge  of  algebra  into 
Europe. 


134.  It  is  often  convenient  to  complete  the  square  without  first 
reducing  the  simplified  equation  to  the  form  in  which  the  coefficient 
of  x^  is  1.     Thus, 

3ar-7       4a;- 10      7 
Example  1.     Solve  — - —  +    ^  _^  ^    =  ^  • 


Process. 

1  =  7. 

3X7-7 

= + 

4  X  7  -  1(1 

7  +  5       " 

--h 

2  +  i-- 

=  i. 

J  = 

--i- 

QUADRATIC  EQUATIONS.  317 

Process.    Clear  of  fractions, 

6  a:2  -  1 4  a:  +  30  X  -  70  +  8  a:2  -  20  ar  =  7  a;2  +  35  a:. 
Transpose  and  unite,  7  ar'^  —  39  a;  =  70. 

Multiply  by  7X4,  196  ar^  -  1092  x  =  1960. 

Add  (39)2,  196  x2  -  1092  x  +  (39)«  =  1960  +  1521  =  3481. 

Extiaet  the  si^uare  root,  14  a:  —  39  =  ±  69. 

Therefore,  14a:  =  39  +  59,  and  14a:  =  39-59.    x  =  7,  and  x  =  -  Y- 
Verify  by  putting  these  numbers  for  x  in  the  original  equation. 

3  X  -  V  -  7      4  X  -  V  -  10  _ 
-1^  +       -Y  +  5       ~*' 

U  -  ¥  =  I. 
i  =  i- 

When  a  quadratic  e({uation  appears  in  the  general  form 

»nx2  +  nx  +  a  =  0, 

the  terms  containing  x  may  be  made  a  complete  square,  without  first 
dividing  the  equation  by  the  coefficient  of  z*.     Thus, 

Transpose  a,  mx^-{-nx  =  —  a 

Multiply  the  equation  by  4  m  and  add  the  square  of  n, 

4m^x^  -\-  4  7nnx  -\-  n^  =  n^  —  4a  m. 

Extract  the  square  root,  2  wi x  4-  n  =  ±  ^/n^  —  4am. 

Transpose  n,  2  w  x  =  — »»  ±  >^n^  ~  4am 

„,       .               —  n  ±  \/w*  —  4am 
Therefore,  x  = .      Hence, 

Hindoo  Method  of  Solving  Quadratics.  Reduce  the  equa- 
tion to  the  form  in  x^  +  n  x  =  —  a.  Multiply  it  hy  f(/icr 
times  the  eoefficient  of  x^,  and  complete  tlu  sqtiare  hy  adding 
to  each  member  the  square  of  the  coefficient  of  x  in  the  given 
equation.  Extract  the  square  root  of  both  member s^  and 
solve  the  resulting  simple  equations. 


318  ELEMENTS  OF  ALGEBRA. 

If  the  coefficient  of  x  in  tlie  given  equation  is  an  even  number,  the 
square  may  be  completed  as  follows  : 

Multiply  the  equation  by  the  coefficient  of  x^,  and  add  to  each  mem- 
ber the  square  of  half  the  coefficient  ofx  in  the  given  equation. 

8x  20 

Example  2.     Solve  —7—0  ""  o~  ==  ^* 
3.  "t"  z       o  X 

Process.     Free  from  fractions, 

(3  x)  (8  x)-20(x  +  2)  =  6  (3  x)  (x  +  2). 
Simplify,  6x^-5Qx  =  40. 

Multiply  by  6,  36  x^  -  336  x  =  240. 

Add  (Af )2,  36  x^  -  336  X  +  (28)'^  =  1024. 

Extract  the  square  root,*  6  a;  -  28  =  ±  32. 

Transpose,  6  a:  =  28  +  32,  or  28  -  32. 

Therefore,  x  =  10,  or  -  f 

Verify  by  substituting  10  for  x  in  the  original  equation. 

8  X  10  20 

Process.  _______  ^  6, 

¥  -1  =  6, 

6  =  6. 
Verify  by  substit-uting  —  f  for  a:  in  the  original  equation. 

8  X  -  f  20 

Process.  _^--^  _____  =  6, 

-¥        20  _ 

-  4  +  10  =  6, 
6  =  6. 
Notes :   1.    *  We  ought  to  write  the  double  sign  before  the  root  of  both 
members.   Thus,  ±  (6  x  -  28)  =  ±  32,  tlie  reason  for  not  doing  so  is  the  same  as 
given  in  Art.  131,  Note. 

2.  The  Hindoo,  or  Indian  Method,  is  supposed  to  have  been  discovered  by 
Aryabhalta,  a  celebrated  Hindoo  mathematician,  and  one  of  the  first  inventors 
of  algebra.  It  is  not  only  more  general  in  form,  but  much  better  adapted  to 
the  solution  of  equations  in  which  the  coefficient  of  the  square  of  the  unknown 
number  is  not  1. 

3.  This  method  has  an  advantage  over  the  common  method  in  avoiding 
fractions  in  completing  the  square,  and  is  often  preferred  in  solving  literal 
equa,tions. 


QUADRATIC  EQUATIONS.  319 

135.  In  case  the  coefficient  of  the  square  of  the  unknown,  in 
the  simplified  etjuation,  is  a  square  number  the  square  may  be 
completed  as  follows  : 

Example  1.    Solve  72  a:  -  54  =  (20  -  z)(4x  +  3). 
Process.     Simplify,  4  x^  —  5  x  =  114. 

""^•^  (i:Jb)' "  6)'  ^  -'  -  ^  - + («' = Mj»- 

Extract  the  root,  2x-  i  =  ±^. 

Transpose,  2  x  =  ^  +  i,a.  or  |  —  ^. 

Therefore,  z  =  6,  or  —  4|. 

The  coefficient  of  x^  may  always  be  made  a  square 
number  by  multiplication  or  division.     Hence, 

General  Method  of  Solving  Quadratics.  Add  to  each 
member  the  square  of  the  quotient  obtained  from  dividing 
the  second  term  by  twice  the  square  root  of  the  first  term. 
Tfien  proceed  as  before. 


^  ,  5         3        35 

Example  2.    Solve  —-r-,  +  - 


X  +  4      X      X  —  2 

Process.     Free  from  fractions, 

5  (x  -  2)  X  +  3  (x  +  4)  (x  -  2)  =  35  (x  +  4)  x. 
Simplify,  -  27  x^  -  1 44  x  =  24. 

Divide  by  -  3,  9  x«  +  48  x  =  -  8. 

(48 X    \* 
— -=  )  ,  or  (8)«,  9  x2  +  48  X  +  (8)2  =  56. 

Extract  the  root,                                          3 x  +  8  =  ±  2  y/\i. 
Transpose,  3x  =  -8±2y'14.     .♦.  x= r-^^^ 


Note.    The  Common  and  Hindoo  Methods  of  completing  the  square  are 
modifications  of  the  General  Method. 


320  ELEMENTS  OF  ALGEBRA. 

Exercise  116. 


Solve  the  following 


1.  23a?=  120  +  a:2;    42 +  ^2^  13  a;;    12  ic^  +  ^  ^  1. 

2.  22a:+ 23-2;2  =  0;    a:2- |rz^=i  32;    2^2  +  3  2;  =  4. 

3.  a;+22-6  2;2=.  0;    25  ^=62:2+ 21;    x2-2x  =  X 

4.  3x'^+12l  =  Ux;  -^i-x=--^-x^;    91  a:2- 2a;  =  45. 

5.  21  ^2  _|.  22  ^  +  5  =  0  ;    9  ^2  -  143  -  G  a;  =  0. 

6.  18a;2-27ir- 26  =  0;    50  a;2  -  15  a;  =  27. 

7.  192;=  15-82;2;     ^2+_4^^^1.     ^2_l^_l3zzO. 

8.  5  2;2  +  14  2;  -  55  ;    (2;  +  1)  (2  2;  +  3)  =  4  2:2  -  22. 

9.  2(^-^)-3(2.'  +  2)(.2;-3);    .32^2  +  2.1  2: +  |  =  0. 
10.  252:+22;2=  42;  |(2;  +  6)(2;-2)  -  f  (62jV +  -V-^)- 

1      __  _1 ^     ^+16       11  _  42: -171 

^^'  IT^       3^^~35'    "T      ■^¥~  3 

42;      x  —  6_4:X+7     '^Jl^       2!  —  2  _  ^ 

^^-  "9"  "^  ^+~3  ~  ~T9~"'    ^~^^  '^  ^^^S  ~  ^^^ 


.0  _J__4^__L_.  _i L-^1- 

3-x       5       9-2  2;'    2^-3       2:+5       18 


14. 


15. 


QUADRATIC   EQUATIONS.  321 

4  3  4  5  3 


X  —  2       X      X  -\-  ^^    X  —  \       x-f2       X 

>y2+  3_  12  +  5a^       'dx         2a;-5_^,^ 
-^^  "^aj2- 5  ~  5(0:^-5)'    2:+  1"^  3a:-l~*^*8- 


^^    5a;-7         a:  -  5        3  a:  -  1       , 
10.  = =  =  -^ zTT^ ;    -^ -—=,  =  1  — 


Ix-b       2a;-13'    4a;+7~  x -\- 1 


,^    12a:8-lla;'-*+  10  a;  -  78       ,,  , 

17.   5-0 — ;, — -T-^ =  l\x  —  h 


3a:+5       3x-5  _  135  7  21  22 

3a;-5       3a;  +  5~176^    x^+'^x^  '6x^-^x~  x 

\  18  7  8  a;  ic  +  3 


19. 


«— 1       a;  +  5      a;H-l       x  —  b'    x  +  %       2a;+l 


136.    Literal   Quadratic   Equations. 

Example  1.     Solve  mx"^  -\-nx  =  — ; —  -^  m x  -  n x^. 
Procesa.    Transpose  and  factor,  (m  4-  n)x^—  {m  —  n)x  =  — —  • 

Multiply  the  equation  by  4(m4-  n)  and  add  the  square  of  (m  —  n)^ 

4  (m  +  nyx^  -  4  (m2  -  n^)  a:  +  (m  -  n)«  =  (m  +  n)«. 

Extract  the  square  root,  2(m-}-n)a:— (m  — n)  =  ±  (to  +  n). 

Transpose,  2  (to  +  n)  a:  =  to  —  n  ±  (m  +  »») 

=  2  TO,  or  —  2  n. 

««       *  TO  n 

Therefore,  x  -       ,      ,  or 


TO  +  n'  TO  +  n 


«      «  ,       2a:+l       1/1       2\       3x4-  1 

Example  2.    Solve  — r (  r  —  ~ )  =  — :: — 

0  x\o      aj  a 


21 


322  ELEMENTS  OF  ALGEBRA. 

Process.     Free  from  fractions, 

ax(2x+l)-  (a-2b)  =  bx(Sx  +  l). 

Simplify  and  transpose, 

2ax^-i-  ax  -3bx^~bx  =  a-~2b. 

Express  the  first  member  in  two  terms, 

(2  a  -  3b)  x^  +  (a  -  b)  X  =  a  -  2b. 

Multiply  by  4  (2  a  -  3  b), 

4  (2a-36)2a;2+4  (2a-36)  (a-b)x  =  8  a^-28ab  +  24  b\ 

Complete  the  square, 
4(2a-36)2a;2+4(2a-36)(a-6)2:  +  (a-&)2  =  9a2-30a6  +  25  62. 

Extract  the  square  root, 

2{2a  -  Zb)  X  +  {a  -  b)  ~  ±  {3  a  -  bb). 
Transpose,  2  (2  a  -  36)  x  =  -  (a  -  6)  ±  (3  a  -  5  6) 

=  2a-4&,  or-2(2a-36. 

a-2b 
Therefore,  x  =  ^^-^g^ ,  or  -  L 

1111 


Examples.     Solve  ^  +  ^"i;-^  =-+ ^  ^  ^ 

111  1 

Process.     Transpose,  -  -  "  =  ^-+6  "  j+^ 

Reduce  each  member  to  a  common  denominator, 

a  —  X  X  —  a 


ax        ia  +  b)(b  +  x) 
Free  from  fractions,  {a-x){a-\-b){b  +  x)=  ax(x- a). 

Transpose  and  factor, 

(a  -  X)  [(a  +  b)(b  +  x)  +  ax']  =  0. 
Hence  (Art.  11),  a  -  a;  =  0.     .',x  =  a. 

Also,  (a-\-b)(b  +  x)  +  ax  =  0. 

Simplify  and  factor, 

^     ^  b(a  +  b) 

b(a  +  b-)  +  (2a  +  b)x  =  0.     .'.  x=  -  ^^^^  ■ 

Kote.    Always  express  the  first  member  of  the  simplified  equation  in  two 
terms,  the  first  term  involving  x'^,  the  second  involving  x. 


QUADRATIC  EQUATIONS.  323 

Exercise  117. 
Solve  the  following  equations  : 

-.       9      /     .  r\      .      7       f\  2  .  ^^^  m  ax 

1.  x^  —  (a-{-b)x  +  ab  =  0:    mx^-] m  =  — 7 — . 

^    X      a      X      b  ^      mbx  am  z 

2.  -  +  -  =  -  +  -;    ma^ m  = r 

a      X      b      X  a  0 

3.  (a--6)r^-(a  + 6)a;+26  =  0;    a^ 3^  ^  abx=^  21?. 

7^      X  ^2a^     2  z(a  —  x)  _  a     a^  +  m2_ 
a^'^  b^~W'     %a-2x  ""V         ^       ~ 

f>,  ^  —  n  X  •\-  'p  X  ^  n'p  ■=■  ^  \    a^  -\-  2x  Vn  =  n. 

6.  a^3^  ~-2a^x+  n*  -1  =  0;    a^ -\-  x  (a  -  b)  =  ab. 

7.  x^-\-mx-^cx-\-Ji^x+m^x  =  0. 

8.  ^a^x  =  (a^-b^  +  xf;    a^  {x  -  a)^  =  1)* (x  +  a)^ 

\a        X  J  \         X        a  J 

10-     —i .     rxQ       =  («  —  ^)^  ;     ^—  ^  X  —        X  =  —CX  —  Z. 

{a-\r  by         ^  '  b  n 

11.  («'~^(^J+l)^2a?;    9a*6*2^»~6a8  63^  =  62. 


19    ^^+  <^  _    ^4-6       11  1        _^ 

«  —  6~~na;  —  a'    a      a  +  a;      a  +  2a7~~ 


324  ELEMENTS  OF   ALGEBRA. 

a-x  X  b      (a-l)^s^  -{-  2(Sa^l)x  _^ 

lo.     -r    ■        " —  —  -  ;      ;;  z —  1. 

X  a  —  X       c  4a  —  1 

14.    ra;  +  ^y  =  4a:2;     (ax-'^=\a?x\ 


137.    Solution  by  a  Formula.     From  the  quadratic  equa- 
tion moi?'  -{■  nx  —  ^  a, 


—  n±  Viv^  —  4:am  ,  ^ . 

*=  2Vv (^> 

By  means  of  this  formula  the  values  of  x,  in  an  equation 
of  the  general  form,  may  be  written  at  once.     Thus, 

Example  L    Solve  10ic2_  23  a:  =  -  12. 
Process.     Here,  m  =  10,  n  =  —  23,  and  a  =  12. 


c  V  .-.  .    .u           1        •    /ix          -(-23)± V(-23)-^-4X  12X  10 
Substitute  these  values  m  (1),  x  = ^  y  10 

23  ±7 
~      20 

=  i  or  f 

^  r.    -,  1  111 

Example  2.    Solve  r- ; —  =  j-  +  -  h 

b  +  c  -\-  y      0       c       y 

Process.     Free  from  fractions,  transpose,  and  factor, 

(h  +  c)y^+  (h  +  cyy  =  -hc{h-[-  c). 
Divide  by  6 +  c,     y^ -^  (h+c)y  = —  he. 
Here,  m  =  1 ,  n  =  6  +  c,  and  a  =  hc. 


Substitute  these  values  m  (1),   x  = ~ 

-(b  +  c)±(b-c) 
~  2 

=  —  r,  or  —  6. 

Note.     In  substituting  the  student  must  pay  particular  attention  to  the 
signs  of  the  coefficients. 


QUADRATIC  EQUATIONS.  325 

Miscellaneous  Exercise  118. 

1.  17  a^  f  19  a*  =  1848;   Sa^  -  12x  +  1  =  6x-2S. 

2.  5a:2  +  4^.  =  273;    ^x'  +  lx  +  ^  =  0, 

^•^+2^=&    «(^'  +  3)«=|(.+  3)^-f 

4        x  +  1  _         a;+2_2a;  +  16      x-2 
'^'^'^bx'^      5      ~^'    x-l~    x  +  5         x  +  1 

2a;+3  7  —  x        7—3x 


5.  16x^-6x'-l  =  0; 


2(2a:-l)      2(a:+l)       4-3a; 


a      5x_^     x-S       2  (a:  +  8)  ^  3  a:  +  10 
3"^'4~3a^a:-5"'"     a:4-4     ~     a:+l 

7.--^=^;    H(5^  +  36)2  =  V^j(8a:2-4)^. 

X  CL 

8.  -4-  =  -  +  -;  lla?»4-10aa:  =  ±a2;  a:' +  -  +  2  = - 
p-\-x      p      x'  'XX 

,-  ,2:2j2.fl,2fltJ 

9.  ax  ■\- dx^  —  a-=  d\    — «  H =  — o  H 

ir^        c        7?r        c 


10.    WX2 


(w^  —  w^  a;  _  a^ a  _ 


mn  '3m  — 2a       2       4a  — (3  w 

11.  a:^  -  2  a  X  =  (6  -  c  +  a)  (6  —  c  —  a). 

12.  x^  -  (a  -{-  b)x  =  \{7n  -\-  n  -\-  a  -\-  h){m  -^  n-a-h). 

,o         1       _^       1  a2  +  a:2      ^^3       ^r  -  3 

a  -k-  X       a  —  X       a*  --  or     x  —  6       x  -\-  .* 


326  ELEMENTS  OF  ALGEBRA. 

14.  ahx'^  -  2x{a  +  h)  ^/'^  =  {a  -hf. 

'  ^'  "^    aW'  ~  18aH2  +         2ah 

a  —  2h~x  5h  —  x         2  a  — x  — 19h_ 

a^  —  4:b^         ax  +  2bx  2bx  —  a  x      ~~    ' 

^^1  1  m  2nx+  n 

^7 I .  =: 

2x^-{-x~-l      2^2— 3  a; +1       2nx  —  n     mx^  —  m 


18.   =  ax. 

ax  —  Va^  x^  —  \       ax  +  \o?  x^  —  1 

Query.  What  is  the  diflference  between  the  meaning  of  "the 
root  of  an  equation"  and  "the  root  of  a  number"? 

138.  Problems.  The  following  problems  lead  to  pure  or  affected 
quadratic  equations  of  one  unknown  number.  In  solving  such  prol)- 
lems,  the  equations  of  conditions  will  have  two  solutions.  Some- 
times both  will  fulfill  the  conditions  of  the  problem ;  but  generally 
one  only  will  be  a  solution. 


Exercise  119. 

1.    Find  a  number  whose  square  diminished  by  119  is 
equal  to  10  times  the  excess  of  the  number  over  8. 

Solution.     Let      x  =  the  number. 
Then,  x  —  8  =  the  excess  of  the  number  over  8. 

Therefore,  a:^  -  119  =  10  (a:  -  8). 
The  solution  of  which  gives,  a:  =  13,  or  —  3. 
Only  the  positive  value  of  x  is  admissible.     Hence,  the  number 
is  13. 

Note.     In  solving  problems  involving  quadratics,  the  student  •should  retain 
only  those  values  for  results  that  will  satisfy  the  conditions  of  the  problem. 


PROBLEMS.  327 

2.  The  difference  of  the  squares  of  two  consecutive 
numbers  is  17.     Find  the  numbers. 

3.  Find  two  numbers  whose  sum  is  9  times  their  differ- 
ence, and  the  difference  of  whose  squares  is  81. 

4.  Find  two  numbers,  such  that  their  product  is  126, 
and  the  quotient  of  the  greater  divided  by  the  less  is  3 J. 

5.  Divide  14  into  two  parts,  such  that  tlie  sum  of  the 
quotients  of  the  greater  divided  by  the  less  and  of  the  less 
by  the  greater  may  be  2r^. 

6.  Find  two  numbers  whose  product  is  m,  and  the 
quotient  of  the  greater  divided  by  the  less  is  n. 

7.  Find  a  number  which  when  increased  by  n  is  equal 
to  m  times  the  reciprocal  of  the  number.  Find  the  num- 
ber when  n  =  17  and  vi  =  60. 

8.  Divide  m  into  two  parts,  so  that  the  sum  of  the  two 
fractions  formed  by  dividing  each  part  by  the  other  may 
be  71.     Solve  when  m  =  35  and  n  =  2^, 

9.  Divide  a  into  two  parts,  so  that  n  times  the  greater 
divided  by  the  less  shall  equal  in  times  the  less  divided  by 
the  greater.     Solve  when  a  =  14,  ?i  =  9,  and  m  =  16. 

10.  A  farmer  bought  some  sheep  for  $72,  and  found 
that  if  he  had  received  6  more  for  the  same  money,  he 
would  have  paid  S 1  less  for  each.  How  many  did  he 
buy? 

11.  If  a  train  travelled  5  miles  an  hour  faster,  it  would 
take  one  hour  less  to  travel  210  miles.  Find  the  rate 
travelled  and  number  of  hours  required. 


328  ELEMENTS  OF  ALGEBRA. 

12.  A  man  travels  108  miles,  and  finds  that  he  could 
have  made  the  journey  in  4-|-  hours  less  had  he  travelled 
2  miles  an  hour  /aster.     Find  the  rate  he  travelled. 

13.  A  number  is  composed  of  two  digits,  the  first  of 
which  exceeds  the  second  by  unity,  and  the  number  itself 
falls  short  of  the  sum  of  the  squares  of  its  digits  by  26. 
Find  the  number. 

14.  A  number  consists  of  two  digits,  whose  sum  is  8 ; 
another  number  is  obtained  by  reversing  the  digits.  If 
the  product  of  the  two  is  1855,  find  the  numbers. 

15.  A  vessel  can  be  filled  by  two  pipes,  running  to- 
gether, in  22|-  minutes ;  the  larger  pipe  can  fill  the  vessel 
in  24  minutes  less  than  the  smaller  one.  Find  the  time 
taken  by  each. 

Solution.  Let  x  =  the  number  of  minutes  it  takes  the  larger  pipe. 
Then,       x  -\-  M  =  the  number  of  minutes  it  takes  the  smaller  pipe. 

-  =r  the  part  filled  by  the  larger  pipe  in  one  minute, 
and  ^^    =  the  part  fi  lied  by  the  smaller  pipe  in  one  minute. 

r^,  P  1  1  1 

Therefore,  -  + 


X  '    a;  f  24  ~  22f 
The  solution  of  which  gives,  x  =  36,  or  —  15. 
One  pipe  will  fill  it  in  36  minutes,  and  the  other  in  1  hour. 

16.  A  vessel  can  be  filled  by  two  pipes,  running  to- 
gether, in  m  minutes ;  the  larger  pipe  can  fill  the  vessel  in 
n  minutes  less  than  the  smaller  one.  Find  the  time  taken 
by  each.     Solve  when  w  =  56  and  w  =  66. 


PROBLEMS.  329 

17.  B  can  do  some  work  in  4  hours  less  time  than  A 
can  do  it,  and  together  they  can  do  it  in  3|  hours.  How 
many  hours  will  it  take  each  alone  to  do  it  ? 

18.  A  boat's  crew  row  7  miles  down  a  river  and  back 
in  1  hour  and  45  minutes.  If  the  current  of  the  river  is 
3  miles  per  hour,  find  the  rate  of  rowing  in  still  water. 

19.  A  boat's  crew  row  a  miles  down  a  river  and  back. 
They  can  row  m  miles  an  liour  in  still  water.  It  took  n 
hours  longer  to  row  against  the  current  than  the  time  to 
row  with  it.  Find  the  rate  of  the  current.  Solve  when 
a  =  5,  w  =  6,  and  n  =  2. 

20.  A  uniform  iron  bar  weighs  m  pounds.  If  it  was  a 
feet  longer  each  foot  would  weigh  n  pounds  less.  Find 
the  length  and  weight  per  foot.  Solve  when  m  =  36, 
a  =  1,  and  n  =  |. 

21.  A  and  B  agree  to  do  some  work  in  a  certain  num- 
ber of  days.  A  lost  m  days  of  the  time  and  received  n 
dollars.  B  lost  a  days  and  received  c  dollars.  Had  A  lost 
a  days  and  B  m  days,  the  amounts  received  would  have 
been  equal.  Find  the  number  of  days  agreed  on  and  the 
daily  wages  of  each.  Solve  when  m  =  4,  71  =  18.75,  a  =  7, 
and  c  =  12. 

22.  A  pei"son  sold  goods  for  vi  dollars,  and  gained  as 
much  per  cent  as  the  goods  cost  him.  Find  the  cost  of 
the  goods.     Solve  when  m  =  144. 

23.  By  selling  goods  for  m  dollars,  I  lose  as  much  per 
cent  as  the  goods  cost  me.  Find  the  cost  of  the  goods. 
Solve  when  m  =  24. 


330  ELEMENTS   OF  ALGEBRA. 


CHAPTEE  XXII. 
EQUATIONS  WHICH  MAY  BE  SOLVED  AS  QUADRATICS. 

139.  In  the  equation  m  {if  —  yy  +  n  (y^  —  z/)^  -f  a  =  0,  suppose 
(y^  —  VY  =  'C,  then  mx^  +  nx  -l-a  =  0.  Similarly,  y^-Sy^  —  9  =  0 
may  be  changed  to  the  form  a:^  —  3  a:  —  9  =  0. 

Hence,  an  equation  is  in  the  quadratic  form  when  the  unknown 
number  is  found  in  two  terms  affected  with  two  exponents,  one  of 
which  is  twice  the  other ;  as,  a;^  +  5  a;^  —  8  =  0. 

The  general  form  for  an  equation  in  the  quadratic  form  is, 

ax^''  +  b^f'  +  c  =  0; 

where  a,  h,  c,  and  n  represent  any  numbers  whatever,  positive  or 
negative,  integral  or  fractional. 

Example  I.     Solve  a;*  -  13  x^  +  36  =  0. 
Process.     Factor,  (x  +  2)  (x  -  2)  (x  +  3)  (ar  -  3)  =  0. 
Hence,   a;  +  2  =  0,   a;-2  =  0,   a;  +  3  =  0,   and  x  —  3  =  0. 
Therefore,  a;  =  ±  2,  or  ±  3. 

Example  2.     Solve  8  a;~  ^  -  15  x~^  -2  =  0. 
Process.     Factor,  (x~^  —  2)  (8  a:~^  +  1)  =  0. 

^-i_2  =  0,  ora:^  =  i         x  =  (hS^      =  i  ^2. 
Also,  8  X   ^  +  1  =  0,  or  a;'  =  -  8.     x=  (-  8)^  =  -  32. 

Example  3.     Solve  3  a:  +  a:^  -  2  =  0. 

Process.     Solve  for  x^.     Thus, 

Multiply  by  12  and  transpose,  36  ar  +  12  a:*  =  24. 

Complete  the  square,  36  a;  4-  12  a:^  +  1  =  25. 

Extract  the  square  root,  6  a:*  +  1  =  ±  5. 

Therefore,  a;*  =  f ,  or  -  L 

Square  each  member,  a:  =  ^,  or  1. 


EQUATIONS  SOLVED  AS  QUADRATICS.  331 

Example  4.    Solve  2  -^^/^^  ^  3  ^^  _  55  _  q 

Process.  Since  -ij/x-^  is  the  same  as  x~ ',  and  /y/x-*^ is  the  same 
as  x~  *,  this  equation  is  in  the  quadratic  form.  Transpose  and  mul- 
tiply by  12,  36  X-*  +  24  x~^  =  672. 

Complete  the  sciuare,  36  z"*  +  24  x~*  +  4  =  676. 

Extract  the  square  root,  6  a:~  *  -f  2  =  ±  26. 

x~^  =  4,  or  —  ^. 

Therefore,  a:'  =  J,  or  —  ^. 

Extract  the  square  root,  a:*  =  db  ^,  or  ±  \/~  ^, 

Raise  to  the  5th  power,*  x  =  ±  jij,  or  ±  V(~A)*- 

Hotes :  1.  When  the  roots  cannot  all  be  obtained  by  completing  the  square, 
the  method  by  factoring  should  be  used.  Thus,  in  solving  a;*  +  7  a:*  —  8  =:  0, 
by  completing  the  square,  we  find  but  two  values  for  x,  re  =  1,  or  —  2.  Fac- 
toring the  first  member,  we  have  (x  +  2) (a*  —  2  x  +  4) (x  - 1 )  {x^  +  x-\-l)  =  0. 
Hence,  a;  +  2  =  0,  a^  _  2  a:  +  4  =  0,  a;  -  1  =  0,  and  x^  -f  a:  +  1  =  0.    Solv- 


ing 


these  equations,  a:  =  —  2,  1  ±  V^^,  1,  and 


2.  •  In  solving  equations  of  the  form  a;»  =  a,  first  extract  the  wth  root, 
and  then  raise  to  the  »ith  power.    In  practice  this  is  the  same  as  affecting  the 


•quation  by  the  exponent  —  .    Thus,  a:  =  a« . 


m 


Example  5.    Solve  a  a:^"  +  2»  jr"  =  —  c. 

Process.   Multiply  by  4  a, 

4  a*  x^'*  +  4a6ar"=:  —  4ac. 
Complete  the  square, 

4a2j:2"  +  4abx'*  +  b^  =  -4ac+b^. 


Extract  the  square  root,           2  a  a*  +  6  =  ±  yb^  —  4ac. 
Transpose  b  and  divide  by  2  a,  af  =  — ^- — • 

Extract  the  -th  root,  x  =  [±VE^ILz*]"  (i) 

Example  6.    Solve  Ax*  -  37 x' +  9  =  0. 


332  ELEMENTS  OF  ALGEBRA. 

Process.     Here,  a  =  4,  6  =  —  37,  c  =  9,  and  n  =  2. 

Substitute  these  values  in(i),  x  =  [^  V(-37)^-4X4X-9-(-37)f 
^^'         L  2X4  J 

r±  35  +  371^ 

=  ±  3,  or  ±  i 

Exercise  120. 

Solve  the  following  equations : 

1.  0^4-14^2^-40;    a;io  + 312^5  =  32;    x^-7x^  =  S. 

2.  2^(19 +  :i^3):^  216;     2^2  +  ^  ^.^  ,,2  +  ^2 

3.  16(':c2+ i^  =  257;    ^3 +14^^1107. 

4.  5  a:*  +  \/x  =  22  ;    'v/^  +  -|-  =  SJ. 

2  V  2- 

5.  ^-t  +  7  ^t  =  44;    3  a;3  +  42  x^  =  3321. 

6.  x^  +  x^=  756  ;    3  V^^  _  4  ^.^  =  7, 


^         .       ..     _2         .         2 

Vx  '       xi 


7.    2V^  +  ^^  =  5;    122:-t  +  f  =  4  +  ¥- 


a  3  ^t  -  a;-|  +  2  =  0  ;    2  a;-5  +  61  a^-t  -  96  =  0. 

9.  x-^  +  ax-i  =  2a^;  x-^-2x-^  =  S',  x-^  +  V^=(^ 

10.  rr*"  -  |2;2»  -  ||  =  0  ;    3  ict'*  +  4  xl""  =  4. 

11.  a:"  +  13  o:''*  =  14  ;  3  :r   '"^  -  26  x   ^'"  =  -16 


EQUATIONS   SOLVED  AS  QUADRATICS.  333 

140.  ?>iuatioijs  may  frequently  be  put  in  the  quadratic  form  by 
grouping  the  terms  containing  the  unknown  number,  so  that  the 
exponent  of  one  group  shall  be  twice  the  exponent  of  the  other  group, 
and  then  solved  for  the  polynomial.     Thus, 


Example  1.    Solve  a;  -  3  x*  -  4  Va:  -  3  x*  -  1  =  -  2. 
The  aquation  may  be  put  in  the   quadratic  form  if  we  reganl 
Vx  —  3  a:*  —  1  as  the  unknown  number.     Thus, 
Process.    Add  —  1  to  each  member, 


x-3a;*-l+4Vx-3ar*-l=-3. 


Put  Vx -3x^-1  =  y,  y^  +  4y  =  -3. 

Therefore,  y  =  3,  or  1. 

Hence,  Vx  -  3  x*  -  1  =  3,  or  1. 

Squaring,  '  x  -  3  x*  —  1  =  9,  or  1. 

Complete  the  square,  x  -  3  x*  +  }  =  ^,  or  Jf . 

Solving  these  equations  for  the  values  of  x,  we  find  x  =  25,  or  4, 

13±3v^ 
and  X  = ^ — — 

Hote  1.  In  solving  equations  of  this  fonn  we  must  group  the  terms  so  that 
the  expression  outside  of  the  radical,  in  the  first  member,  is  the  same  or  a  mul- 
tiple of  the  expression  under  the  radical  sign. 

Example  2.     Solve  x*  -  6  ax«  +  7  a" x^  4-  6  a»x  =  24  a*. 

Process.  Add  2a«x^     x*-6ax^+9a^x^-^6a*x  =  24  a*  -f  2a«x«. 
Transpose  2 a«x2,  x*-  6  a x«+  9  a^ x2+  6 a«x  -  2  a^x^  =  24  a*. 
Group  and  factor  the  terms, 

(x2  -  3  ax)2  -  2  a2  (x2  -  3  ax)  =  24  a*. 
Regard  x*— 3  ax  as  the  unknown  number,  and  complete  the  square, 

(x2  -  3  a  x)a  -  2  a«  (x2  -  3  a  x)  -f-  a*  =  25  a*. 
Extract  the  square  root,  (x'^  —  3  a  x)  —  a^  =  ±  5  a* 

Therefore,  x«  -  3  a  x  =  6  a",  or  ~  4  a«. 

Complete  the  square  and  solve,  x  =  2  (3  ±  \/33), 


334  ELEMENTS  OF   ALGEBRA. 

Note  2.  Form  a  perfect  square  with  xi  and  —6a  x^.  The  third  terra  of  the 
square  is  the  square  of  the  quotient  obtained  by  dividing  6ax^  by  twice  the 
square  root  of  x*. 

Example   3.     Solve  a:2  +  4x  —  4a:r-i-fx-2  =  ^. 
Process.     Use  positive  exponents,  rearrange  terms,  and  factor, 
a=^  +  ^,  +  4(x-i)=|. 

Regard  x as  the  unknown  number,  and  subtract  2  from  both 

sides,  „      ^       1        .  f        1\  11 

Factor,  and  complete  the  square, 


(.-iy+4(.-^)+4=^. 


Extract  the  square  root,  a:  —  -  +  2  =  ±f. 

Therefore,  x  —  -  =  —  ^,  ot  —  ^. 


X 


Free  from  fractions,  ^  ar^  —  1  =  —  ^  a:,  or  —  ^  x. 

Complete  the  square  and  solve,  x  =  ^  (—  1  ±  y'S?) , 

x  =  ^(-ll±^/T57). 

Note  3.    Form  a  perfect  square  with  x^  for  the  first  term  and  —  for  the 

third.    The  middle  term  will  be  twice  the  product  of  their  square  roots  taken 
with  a  negative  sign. 

A  Biquadratic  Equation  is  an  equation  of  the  fourth  de- 
gree. Biquadratic  means  twice  squared,  and  hence  the 
fourth  power. 

If  a  biquadratic  is  in  the  form, 

x^+2mx^+  (m2  +  2  n)  a:^  +  2  mnx  =  a  (ii) 

the  first  member  becomes  a  perfect  square  by 

Adding  n^,  or  the  square  of  the  quotient  obtained  by  dividing  the 
coefficient  of  x  by  the  coefficient  of  x^. 


EQUATIONS  SOLVED  AS  QUADRATICS.  335 

Thus,  extracting  the  stiuare  root  of  the  fii-st  member, 

X*  i-  2mx^  +  {inr  +'2,n)x^  -\-  2mnx  |  x'^  +  mx  -{-  n 

X* 

2x^-\-  mx  I  2mx^+  (m^  +  2n)x^-\-2mnx 

2mx*+{m^  )x^ 

2x^  +  2mx-\-n\  +  (  2n)x^  +  2mnx 

+  (  2n)x^+  2mnx  +  n^ 

—  n*.     Hence, 
the  equation  may  be  written, 

{x^  -\-mx  +  n)2  -  n^  =  a,  or  (z"  +  w  z  +  «)«  =  a  +  n^     (iii) 

Example  4.     Solve  x*  -  10  ««  +  35  z^  _  50  x  =  1 1. 

Process.    Here,  2m  = —  10,  2 win  =  —50.     .-.  m  =  — 5  and  n  =  5. 

Since  m^  +  2n  =  35,  the  equation  has  the  form  of  (ii). 

Add  25  ;  or  put  w  =  —  5,  n  =  5,  and  a  =  1 1  in  (iii), 

(z2  -  5  z  +  5)2  =r  36. 

Extract  the  square  ^oot,  z*  —  5z  +  5  =  ±6. 

Therefore,  z^  -  5  z  =  1,  or  -  11. 

5  ±  a/29 
Complete  the  square  and  solve,  z  = ^ —  , 

5±  a/^Hq 
^  = 2—- 

Kote  4.  After  adding  the  value  for  n*  the  first  member  may  be  factored  by 
substituting  the  values  for  m  and  n  in  (iii). 


Exercise  121. 
Solve  the  following  equations  : 

1.   (3^  +  x-'2)^-lS(2^  +  x-2)  +  Z6  =  0. 


2.  a^»  4-  2  a;  +  6  Va^+2x+5  =  11. 

3.  2:2  +  24  =  12  V^-qrf.    2a:  +  17  =  9  V2  a;  -  1. 

4.  a:2  -  a:  +  5  (2a:2  -  5  a;  +  6)i  =  J  (3  a:  +  33), 


336  ELEMENTS   OF  ALGEBRA. 

^  (a;2  _^  ^,.  ^  6)i  ^  20  -  |(a;2  +  a;  +  6)^ 

7.  («  +  l)%4(«+!)  =  I2. 

8.  (a;2  -  5  a;)2  -  8  (ic2  _  5  ^)  ^  43^ 

9.  9  ^  -  3  ^2  _!_  4  (^2  _  3  ^  +  5)^  ^  il^ 

■«(-9"-!(-^)-S- 

11.  (3a:2-10^+ 5)2-8(3^- 10a^+ 5)  =  9. 

13.  :i:4  +  6^^  +  5:z;2-  12a;=12. 

14.  x^-Qa^-2^x^+  114  2;  =  80. 

15.  ^4  +  2  2^  -  25  a;2  _  26  a;  +  120  =  0. 

16.  2:4-8  .>;3  +  10  2:2  +  24  a:  +  5  =  0. 

17.  a:4  +  8  2:3  +  2  a:2  -  I  a:  =  |_ 

18.  (^3  _  16)1  _  3  (2^3  -  16)i  =  4. 

19.  ^-Y^^-Vx-x-^^A.;  2;2+3:^-32:-i  +  2r2  =  ^. 


EQUATIONS  SOLVED  AS  QUADRATICS.  337 

141.   Equations  Containing  Radicals  may  be  Solved.   Thus, 

Example  1.    Solve  x  -  ^3*  +  2  x  +  12  +  2  =  0. 

Process.     Transpose,  x  +  2-  ^x»  +  2x  -f-  12. 

Raise  both  members  to  the  third  power, 

a:»  +  6  x-»  -f  12  X  +  8  =  a:*  +  2  X  -h  12. 
Transpose  and  simplify,  3  a:"^  +  5  x  —  2  =  0. 

Factor  and  solve,  x  =  ^,  or  —  2. 

Verify  by  putting  these  numbers  for  x  in  the  original  equation. 

Process,    x  =  J.  x  = 

2  -  ^-8-4+12  +  2  =  0, 

-2-0  +  2  =  0, 

0  =  0, 

1  1 


i  -  ^tjV  +  f  +  1^  +  2  =  0, 

i  -  i  +  2  =  0, 

0  =  0. 


Example  2.    Solve 


Vx*  +  1      ^/x'^  -  1      ^/J^  -  1 


Process.     Multiply  by  /^/x*  —  1, 


Vx2  -  I  +  yx«  +1  =  1. 
Square,  x^  -  1  +  2  ^/x'^~^\  +  x«  +  1  =  1. 

Transpose  and  simplify,  2  ^/x*  —  1  =  1  —  2  x^. 

Square,  4a:*-4  =  l-4x«  +  4x*. 

Simplify,  x«  =  f . 

Extract  the  square  root,  x  =  i  J  y'S. 


Exercise  122. 
Solve  the  following  equations : 


1.  3  V^+6  +  2  =  a;+VT+6;    a:+  \/iT~2  =  10. 

2.  aj  4-  16  -  7  v./-  +  16  =10-4  Vx  +  16. 

3.  2a:+  V4a;  +  8  =  J;    V4a;+  17+  V^l  »  4. 


338  ELEMENTS   OF  ALGEBRA. 


4.   2'v/3a;  +  7  =  9-V2i^-3 


-     V4:X  +  2       4  -  Vi 


4A/ic  Vx 


12  5x-  9    _         \/5^-3 

Vic+  12'    V5^+3  2 

44-2; 


5.  V^  +  ^  =    y  ;   -7= — ;==!  + 

Vic  +  12      V  0  a;  +  3 

6.  Vi^-2\/^-=2;;    1^64  -{-2x^-Sx-    ,, 

V4:  -\-  X 

H,    »  A/       .      *  A/      o        /o—       3.a;  —  1       .   ,  Vi^a;  — 1 


+ ^  =  a;. 


X  +  V'2  —  x^      X  —  V2  —  x^ 


^    V7  y2  +  4  +  2  a/3  7/  -  1       ^     m-  Vrn^  -  y^ 
a/7  2/^  +  4  -  2  A/3y-  1         '   ^^^  +  A/m2  -  2/2 

10.    ^a2  +  2a^2_2aaj  =  -^^iL;    A/6a:-a;3=     "^ 


11. 


Va  +  .^'  A/a; 

a^-&2        V^  +  &      a/.^:  +  9       3a/^-3.8 


a/^  2;  +  &  ^         '        Vx  9  —  Vic 

^^       /— --         / 12a  6  +  SVx       4 

12.  Va  +  X -{■  ya  —  x= ;    7— r — 7=^  =  "7= 

5  A/a  +  a;      4  +  Y^,^         Va; 

13.  V^f^+Jj  -  Vy^^  =  V2y;    2rr+3A/^=27. 

,  ,    a;  +  a/S       2:2-2;  12  +  8  a/S 

14    ^  —  —  —  •    X  — -z 

X-  Vx  4      '  x-b 

o_^  +  ^'^^.    a;  _  A/a;  —  12 
2;        '4         2:  —  18 


16.   A/2;^  +  A/2r^  =  6  a/5;    Vx  —  a  ■\-  "sJx -^  a  =  ^/'^ 


THEORY  OF   QUADRATIC  EQUATIONS.  339 


THEORY  OF  QUADRATIC  EQUATIONS. 

142.    Representing  the  roots  of  mx^  +  nx  =  ~  a  by  r  and  r^,  we 
have  (Art.  134), 


—  n  +  \/n^  —  4  a  m  ^ 

—  n  -  \^n^  —  4am 

'1-                'Zm 

n 

'  +  '.  =  -» 

(i) 

a 

rr,  —  — 

*■      m 

(ii) 

Adding, 

Multiplying, 

Hence,  if  a  quadrntic  appears  in  the  form,  mx^  +  nx  =  —  a^ 

I.  The  sum  of  the  roots  is  equal  to  the  quotient ^  with  its  sign  changed, 
obtained  by  dividing  the  coefficient  ofx  by  the  coefficient  ofx^. 

II.  The  product  of  the  roots  is  equal  to  the  second  member,  with 
its  sign  changed,  divided  by  the  coefficient  of  x^. 

By  means  of  (i)  and  (ii)  the  ori«,Mnal  equation  becomes, 

m  x*  —  wi  (r  -f  r,)  a:  4-  m  r  Tj  =  0  (1) 

Factor,                              m(x-r)(x  -  r,)  =  0  (2) 
If  m  =  1,                                           x^-\-nx  =  —  a 

x^-  (r  +  rj)  X  +  rrj  =  0  (iii) 

ix-r)(x-r,)  =  0  (3) 

If  the  roots  of  a  quadratic  equation  be  given,  by  means 
of  (iii)  we  can  readily  form  the  equation. 

Example  1.     Form  the  equation  whose  roots  are  ^,  —\. 
Process.     Here,  r  =  ^  and  r^  =  -~  \. 

Substitute  these  values  in  (iii),  x^-  (^-\)x  +  (\)  (-\)  =  0. 
Simplify,  8  x«  -  2  x  -  1  =  0. 

Example  2.    Find  the  sum  and  the  product  of  the  roots  of 
8x«+3x-5  =  0. 

Process.     Here,  m  =  8,  n  =  3,  and  a  =  —  6. 

Substitute  in  (i)  and  (ii),    r  +  r j  =  —  {  and  rrj  =  —  j. 


340  ELEMENTS  OF  ALGEBRA. 

Exercise  123. 

Find  the  sum  and  product  of  the  roots  of : 

1.  2:2  +  8^  =  9;    12  a:'^  -  187  ^"  +  588  =  0. 

2.  20x-^  =  5-5x-^l    a^-6x+9  =  9x. 

3.  32:2  + 5  =  0;    ^24.^2:=^^.    a^-l5x  =  S. 

.      „      2mn^x  mn         207,  2  .   1.2      n 

4.  x^ = ;    x^  —  2  0  X  —  0?  -\-  h^^  =  0. 

m  —  n  m  —  n 

Form  the  equations  whose  roots  are : 

5.  7,-3;    |,-|;    5, -3;  ±  V=^;  2- V3,  2  +  V3. 

6.  0,-5;   7  +  2A/5,  7-2a/5;    1  +  V2,  1  -  V2. 

7.  7/1  (m  +  1),  1  —  //I ;    — , ; 1-  ,  0  —  a. 

^  ^  n         m       a  —  h 

8.  -  w  +  2  \/2  /I,  -  w  -  2  V2  71,  >  -^— 

143.  A  Root  is  said  to  be  a  Surd  when  it  can  be  found 
only  approximately  ;    as,  a;  =  =t  ^^, 

Real  Roots  are  values  of  the  unknown  numbers  that  can 
be  found  either  exactly  or  approximately. 

Imaginary  Roots  are  values  of  the  unknown  numbers 
that  cannot  be  found  exactly  or  approximately;  as, 

x  =  ±  V^^. 
Character  of  Roots.      For  brevity,   represent  the  roots  of  the 
equation  mx^-^nx  +  a- 0  by  r  and  r^,  then, 

r= TJ}L , 

2m 

_  —  n  —  \/7i^  —4  am 

^^  2m  "' 


THEORY  OF  QUADRATIC  EQUATIONS.      341 


It  is  seen  that  the  two  roots  have  the  same  expression,  y/n^—Aam. 

If  n^  is  greater  than  4  am,  n*  —  4  a  m  will  be  positive^  and 
\/n*  -4am  can  be  found  exactly  or  approximately. 

If  n  is  positive,  r^  is  numerically  greater  than  r ;  if  n  is  negative, 
r  is  numerically  greater  than  Tj.     Hence, 

I.   Condition  for   Eeal   and   Different  Ebots.     n*  -  4  a  m, 

positive. 

nitiatration.  3a:2-2x  +  |  =  0. 

Here,  wi  =  3,  n  =  —  2,  and  a  =  |. 

n2-4am=  (-2)2 -4X^X3=4-1  =  {. 

Therefore,  the  roots  are  real  and  different. 

Evidently  both  roots  will  be  rational  or  both  surds  according  as 
n*  —  4  a  m  is,  or  is  not,  a  perfect  square.     Hence, 

11/  Condition  for  a  Rational  or  a  Surd  Root,     n^-  4am, 

a  square  number;  or,   /y/n-  — 4a7/t,  a  surd. 

lUustrationB.     (1)  z*  -  3  x  -  4  =  0 ;    (2)  8  x^  +  5  x  -  J  =  0. 

(1)  Here,  m  =  I,  n  =  -  3,  and  a  =  —  4. 
n2  -  4  am  =  (-  3)*  -  4  X  -  4  X  1  =  9  -H  16  =  25. 

Therefore,  the  roots  are  real  and  rational,  and  dilferent. 

(2)  Here,  m  =  8,  «  =  5,  and  a  —  —  \. 

Vw*  -4am  =  \/25  +  8  =  \/33- 
Therefore,  the  roots  are  real  and  surds,  and  different. 


If  n*  is  less  than  4am,  n'— 4am  will  be  negative,  and  \/«*— 4am 
will  represent  the  even  root  of  a  negative  number.     Hence, 

III.   Condition  for  Imaginary  Roots.     n«-4am,  negative, 

Uluatration.  2x«-3x  +  2  =  0. 

Here,  m  —  2,  n  =  —  3,  and  a  =  2. 

n«-4am=(-3)« -4X2X2^9-  16  =  -7. 
Therefore,  the  root.<<  are  both  iniM^'inary. 

If  n*  =  4am,  n*-4«m  =  0,  and  the  roots  will  be  real  and  equals 
and  have  the  same  sign,  but  opposite  to  that  of  n.     Hence, 


342  ELEMENTS   OF   ALGEBRA. 

IV.  Condition  for  Equal  Roots,     n^  ~4am  =  0. 

Illustration.  4x2  —  12a;  +  9  =  0. 

Here,  m  =  4,  n  =  —  12,  and  a  =  9. 

71^ -4  am  =144-  144  =  0. 

Therefore,  the  roots  are  real  and  equal. 

If  a m  is  positive,  for  real  roots,  n^  —  4am  will  be  positive  and 
less  than  n^,  since  ^n^  —  4am  will  be  less  than  n. 

If  a 771  is  negative,  ^n^  —  4  am  will  be  greater  than  w,  since 
n^  —  4  a  m  will  be  greater  than  n^.     Hence, 

V.  Condition  for  Signs.  //  a  m  is  positive,  real  roots  have  the 
same  sign  but  opposite  to  that  of  n.  If  am.  is  negative,  the  roots 
have  opposite  signs. 

Illustrations.     (1)  2x2-10a;+12  =  0;    (2)  2x^-5x-3\^  =  0. 

(1)  Here,  m  =  2,  n  =  -  10,  and  a=  12. 

n2  -  4  a  m  =  100  -  96  =  4. 
Therefore,  the  roots  are  rational  and  positive,  and  different. 

(2)  Here,  m  =  3,  n  =  —  5,  and  a  =  —  S^J. 

n2 -4  am  =  25  +  47  =  72. 
The  roots  are  surds  and  have  opposite  signs,  and  different. 


Exercise  124, 

Determine  by  inspection  the  character  of  the  roots  of : 

1.  5x'^-x  =  3;    7x^+2x=-^:    Ux^-x^-^, 

2.  4^2+ 52a;  =  87;    3  x^  +  4a' +  4  =  0. 

3.  6-llx-9x'^  =  0;    9a  =  3  +  4la;2 

4.  10x+S:^  =  -3x^-    lx'^-^x  +  l=0. 

5.  3a:2-2a:+3=:0;    4:X^-3x-5  =  0. 


THEORY  OF  QUADRATIC  EQUATIONS.  343 

8.  6ar2+ 5  2:-21  =  0;     13  ar^  +  56  a;  -  605  ==  0. 

9.  9  ar^  -  30  a;  +  41  =  0 ;    40  o,^  -  100  a;  -  360  =  0. 

Query.     How  many  roots  can  a  quadratic  equation  have  ?  Why  ? 

Miscellaneous  Exercise  125. 
Solve  the  following  equations : 
1.   a:t  +  7J  =  44;   x-^-2x-^  =  S;   3s^-:^-2  =  0, 


2.  k/t-^-  +  J- =  2X-     1  +  8  a:*  -h  9  '>y^  =  0. 

▼   i  —  X        ^       X 

3.  —==  —  2  V2  a:  =  59  ;    .,  ^  ,     ._  +  — =  =  3  Va;. 
^/2x  1  +  5  Va;       Vx 

4.  a;*"-2a:3«  +  ^  =  6.    3^-2o(^  +  x  =  a. 

5.  a^  +  -^a:3_39^  =  81.    a^  _  2  a^  +  a;  =  380. 

6.  108a:*=  180^8- 20aj- Sla^H- 7. 

7.  a:*-10a:2   +  35  a:^  _  50^;  =  -  24. 

8.  {x  -  rt)f  +  2  v^  (a;  ~  rt)i  -  3  7i  =  0. 

9.  a:t  ~  4  a:f  +  r-  J  +  4  a:-f  =  -  {. 


344  ELEMENTS   OF   ALGEBRA. 


V  y  -\-  2  a  —  ^y  —  2  a  _    y       x  -\-  ^x  _  o^  —  x 
^y-2a  +  V2/  +  2a  ~"  2  a'    x  -  '^x  4 


4:r" 


11.    3a;"'^a;"-^:=:4;  V'6:r+l  +  K2:  +  4+ V^^+l  =  2. 


12.  a;  V5  +  a/2  a'  4-  2  =  V^  +  :^;    a:  -  1  =  2  +  — -  • 

13.  2  Vi  +  2  :z^-i  =  5  ;    6  Vi  =  5  :c-  2  -  13. 

14.  x^  +  2  m^x-^  =  ^m;    oT^  +  2  =  ^  ,"^      • 

X-   3  +  5 

15.  ^^2;+  V2^^=T-Kx- V2^T  =  I v/ — ^^^=. 


ic 


+  Va^  -  1        a:  -  V:r2  -  1 


16.  ^^  ^-^  -  ::       ^  =  8  ^'  Va:^  -  3  ^^  +  2. 

a;  —  Vx^  —  1       2:  +  Va;2  —  1 


17.  — ^^ — ^~^, ^^-—  =  & ;    ic3  +  a;  A/a;  -  72  =  0. 

ft  +  2; 

18.  State  the  conditions  that  will  make  the  roots  of 
x^  +  Ax  +  B  =  0:  (i)  surds;  (ii)  real;  (iii)  imaginary; 
(iv)  equal;  (v)  have  same  signs;  (vi)  have  opposite  signs; 
(vii)  equal  in  value  but  opposite  in  sign. 

19.  Find  a  number  such  that  if  its  nth  root  be  increased 
by  one  half  of  its  ^th  root,  the  sum  shall  be  a.  Solve 
when  n  =  2  and  a  =  5. 

20.  Find  a  number  sucli  that  if  its  nth  power  be  dimin- 

2  a  .         ' 

ished  by  the  -  th  root  of  the     th  part  of  it,  the  remainder 
-'         n  c       ^ 

shall  be  m.    Solve  when  m  =  144,  n  =  2,  a  =  27,  and  c  =  5. 


SIMULTANEOUS  QUADRATIC  EQUATIONS.  345 


CHAPTER  XXIII. 


SIMULTANEOUS  QUADRATIC  EQUATIONS. 

144.  Only  certain  forms  of  quadratic  equations  involving  two 
unknown  numbers  can  be  solved.     Thus, 

Example  1.    Solve  the  equations :  ^  ^  ^„    ^~  «  «      ^.         ^2 

10  -  1/ 
ProcesB.     From  (1),  x=      ^  (3) 

/l()_y\2      /10-w\ 

Substitute  in  (2),  2  (  — ^-^  j  -  ( "^ )  2/  +  3  3^«  =  54. 

Simplify  and  factor,  (y  —  4)  (4y  +  1)  =  0. 

Therefore,  y  =  4,  or  —  ^. 

Substitute  in  (3),  ar  =  3,  or  5^.     Hence, 

When  one  of  the  Equations  is  of  the  First  Degree.  Solve 
by  substitutiou. 

The  Degree  of  a  term  is  the  mimber  of  literal  factors 
involved,  and  is  always  equal  to  the  sum  of  their  ex- 
ponents. 

Each  literal  factor  is  called  a  Dimension. 

Thus,  3  xy  is  of  the  second  degree,  and  has  two  dimensions. 
5  x«j/*  is  of  the  fjih  degree,  and  has  Jive  dimensions. 

«    o  1      .1-  *•         5l83xy  +  72x+36y  =  88     (1) 

Examples.  Solve  theequations :  |  ^^^^^^^^^^3^^  ^  g^    ^^^ 

80-36y        .^x 
Process.     From  (2),  a;  -  Y77— jTgo       ^'  ^ 

Substitute  in  (1), 

183(80y-36y«)      72(8<l-36y)  ,  _„  _  ^^ 
177y  +  60        +      177  y  4-60     +  ^^  ^  "  ^^^ 
Simplify,  9  y*  +  57  y  -  20  =  0. 

Complete  the  square  and  solve,  y  =  J»  or  —  6f . 

Substitute  in  (3),  ar  =  ^,  or  -  f 


346  ELEMENTS  OF  ALGEBRA. 

Example  3.     Solve  the  equations  .  |  6  x^  -  x  -  3  3/  =  5  (1) 

^  lx^  +  x-y=l  (2) 

Process.     From  (1),  y  = :: (3) 

Substitute  in  (2)  and  simplify, 

3a:2-4a:-2  =  0.  _ 

Complete  the  square  and  solve,  x  = :z 

^  ,     .         .     ,  .                                               11  ±7  ViO      ,, 
Substitute  m  (3),  y  — ^ .     Hence, 

When  each  Equation  Contains  only  one  Second  Degree 
Term,  and  that  Term  Consists  of  the  Same  Product  or  Square 
of  the  Unknown  Numbers.     Solve  by  substitution. 

Exercise  126. 

Solve  the  following  equations  : 

C^xy  =  50.  C2x  +  y  =  22. 

2.    |3.;j7/+6^-2z/  =  4.     g      {x-y  =  ^. 
\4.xy  —  X  +  ^y  =^l.        '    \xy=126. 

^      (x  +  xy  =  24.  ^Q      r  2^3 -7/ =  218. 

'    \xy  +  y  =  21.  '    \x~y  =  2. 

4      f2y2  +  y:=28.  ^^      fx--y=:4. 

^      (15  +  y  =  x.  ^2      (x  +  3y=16. 

\xy  =  2ij^.  '    \3x^-h2xy-y^  =  -12 

Q^     (xy+Qx  +  7y  =  66.    ^3      {frJ^  +  y2='iS5. 
\Sxy  +  2x+5v  =  70.        '     lx-v=3. 


SIMULTANEOUS  QUADRATIC   EQUATIONS.         347 
15/2^  +  ^  =  9.  .Q     (x-y  =  4. 

145.  All  equation  containing  two  unknown  numbers  is 
symmetrical  when  the  unknown  numbers  can  change  places 
without  changing  the  equation ;  q^q, 'i  a?  —  4:  x  y  ■\- Z 1/  =  2  \ 
ic*  4-  b  3^y  -\-  5  xij'^  +  y^  =  —  5  x^  ?/. 


Example  1.     Solve  the  equations : 

(  x^  +  ,/  =  89                    (1) 
Uy  =  40                            (2) 

Process.     Add  (1)  to  twice  (2), 
Siibtract  twice  (2)  from  (1), 
Extract  the  square  root  0!  (3), 
Extract  the  square  root  of  (4), 

x2  +  2xi/-f  2/2^  169     (3) 
x2-2xy  +  y2-9         (4) 

x  +  y  =  ±  13. 
X  -  1/  =  ±  3. 

We  now  have  to  solve  the  four  pairs 

of  simultaneous  equations, 

ar-f  t/=13>      x  +  y=  13    >      x -f  y 
X-  y=    3r    x-y  =  -zy    x-y 

r^--13)      x+y  =  -137 
=  3       y    x-y  =  -    3>' 

There  are  four  pairs  of  values,  two  of  which  are  given  by  x  =  Jb  8, 
'/  =  ±  5,  and  the  other  two  by  x  =  ±  5,  y  =  ±  8,  in  which  the  upper 
signs  are  to  be  taken  together,  and  the  lower  signs  are  to  be  taken 


together. 


Kotes :  1 .  If  the  second  members  of  two  simple  equations  have  the  sign  ± , 
we  will  have  six  simultaneous  .simple  equations  to  consider. 

2.  The  above  equations  may  be  solved  as  in  Art.  144,  but  the  symmetrical 
nwtliod  is  more  simple. 

K\.\Mi'LE  2.     Solve  the  equations:  \    „      ^~      „      _,  \J. 

^  lx^-xy  +  y^-2l  (2) 

Process.     Divide  (1)  by  (2),  x  +  ?/  =  6  (3) 

Square  (3),  x^  +  2xy-\-y^  =  36  (4) 

Subtract  (2)  from  (4),  3xy=  lb,  or  xy  =  b  (6) 

Subtract  (5)  from  (2).  x^  -  2  x.V  +  y2  =  16. 

Extract  the  square  root,  x  —  y  =  ±  4  (6) 

Add  (3)  and  (6)  and  divide  the  result  by  2,  x  =  5,  or  1, 

Subtract  (6)  from  (3)  and  divide  the  result  by  2,  y  =  1,  or  5. 


348  ELEMENTS  OE  ALGEBRA. 

Example  3.     Solve  the  equations  .  5  ^'^  +  2/^  -  ^  -  2/  =  "8       (1) 
^  ixy  +  x  +  y  =  2^  (2) 

Process.    Add  (1)  to  twice  (2),  x'^+2xy  +  y'^+x  +  ij  =  156. 
Factor,  {x  +  yy  +  (x  +  y)  —  156. 

Regard  a;  +  ^  as  the  unknown  number,  complete  the  square,  and 
solve,  *  a;  +  ?/  =  12,  or-  13  (3) 

Subtract  twice  (2)  from  (1),  factor,  and  transpose  ^(x  +  y), 

{X  -  3,)2  =  3  (X  +  y-)  (4) 

From  (3)  and  (4),  {x  -  yY  =  36,  or  -  39. 

Therefore,  x  -  y  =  ±  6,  ov  ±  y.1~39  (5) 

-  13  ±  V-39 
Add  (5)  and  (3),  etc.,  a;  =  9,  or  3,  and ^ 

Subtract  (5)  from  (3),  etc.,  y  =  3,  or  9,  and -^ • 

Example  4.     Solve  the  equations :  <  ^  ^  ^ 

^  lx  +  y  =  4  (2) 

Process.     Raise  (2)  to  the  fourth  power, 

x*-\-4x^y  +  ex^y^+ 4xy^  +  y^=256  (3) 

Subtract  (1)  from  (3),  etc., 

2x»y  +  3x^y^+2xy^-  87  (4) 

Square  (2)  and  multiply  the  result  by  2  xy, 

2x»y-h4x^y^-^2xy»  =  32xy  (5) 

Subtract  (4)  from  (5),  etc.,  x^y^-  32xy  =  -  87. 

Regard  xy  a.^  the  unknown  number,  complete  the  square,  and 
solve,  xy  =  29,  OT  3. 

We  now  have  the  two  pairs  of  equations  to  solve, 
X  -^y=    4\  X  +  y=  4) 

xy=29)  '  _xy  =  3} 

^  ar  =  2  ±  5  \/^> 

From  the  first,       A  "^  ^ 

l2^^2T5V-l. 


(0;  =  3,  or  1. 

'  \y=\,  or 3. 


From  the  second,   ^ 

When  the  Equations  are  Sjrmmetrical     Combine  them  in 
such  a  manner  as  to  remove  the  highest  powers  of  x  and  y. 


SIMULTANEOUS  QUADRATIC  EQUATIONS.  349 

Exercise  127. 
Solve  the  followiug  equations  : 

'    \x  +  7j  =  n.  '    \x^-xy  +  y^=2l. 

^      ix^  +  x  +  i/+i/=lS.        g      (2^+x^y^+y^  =  9Sl. 
'    \xy  =  6.  '    \x^  +  X  y -\- i/  =  49. 

^      (x^y^+2x+2y=50.  ^Q      (x^-xy  +  y^=U. 
\xy  +  x  +  y  =  2d.  '    \x  +  y=U. 

g      (2^  +  yi  =  52.  j^      ra^4-a^»v/2  +  /=133. 

\  X  +  y  +  xy  =  34.  '    \  o^  +  x  y  +  y^  =  19. 


6.    <^  2:2  +  2r^  -  900  •  12.    ^  a?»  +  2r*  ~  ^  * 

l«y  =  30.  U  +  y  =  8. 

146.     An  algebraic  expression  is  said  to  be  honwgeneovs 
when  all  its  terms  are  of  the  same  degree. 

Thus,  9  x*<>  -f  3  X  y*  —  8  r*  y*  is  homogeneous,  for  each  term  is  of 
the  10th  degree  and  has  ten  dimensions. 

Example  1.    Solve  the  equations  H  «  «     «  „   «     «         \^i 

Process.     Let  y  —  vx,  .iiifl  substitute  in  both  equations. 
From  (I),  6x2-f- 2t'«z«-6t;x«=  12. 

12 
Therefore,  ^'=  6-5»; +  2  »;«     ^^^ 

From  (2),  3x«  +  2yx2  =  3  r^x^*  -  3. 

Therefore,  ^'  =  3t;«-2t;-3     ^"^^ 

12  3 

Equate  (3)  and  (4).  e-5v  +  2t^  =  3ra-2t;-3 

Simplify  and  solve  for  »,  r  =  },  or  - 1. 


350  ELEMENTS   OF  ALGEBRA. 


Substitute  y  =  | 


12 

=  4. 


6-5  X  1  +  2(1)' 


Substitute    v  =  —  ^  in  (3), 

12  25 

"^        6-5X-|  +  2(-f)2-3l' 
.-.       x  =  i  /j  V31. 
3/  =  -f^-T^\V31. 


Notes :  1.  In  finding  the  last  values  of  x  and  y,  it  will  be  observed  that  ± 
values  of  x  gives  respectively  —  and  +  values  of  y.  This  indicates  that  the 
equations  can  be  satisfied  only  by  making  y  ^  —  ^^  V2>1,  when  x^-\-  ^^^  V'6\ ; 
and  when  x  =.  —  ^^  V31,  y  must  be  +  ^f  VZl. 

2.    The  sign  T  denotes  precedence  of  the  negative  value. 

When  each  Equation  is  of  the  Second  Degree  and  Homo- 
geneous.    Substitute  v  x  tor  y  in  botli  equations. 

Exercise  128. 

Solve  the  following  equations  : 

^    ^  x^  +  xy  =  lb.  n     (  x^—  o  xy  +  y'^  =  —  1. 

'  \y^  +  xy=  10.  '  \Sx^-xy  +  Sy'^  =  lS. 

2     (x''-xy  =  24:.  ^    (2x^-5xy+3y'^=l. 

\  X y  —  y^  —  8.  *  \  3  x^—5  xy+'2y'^  =  4=. 

=  21. 
18. 


2     (x^  +  4:Xy=lS3.  g     (x^~2xy 

'  \4:xy  +  16/ =  228.  *   I  xy  +  y^  = 

^    (2x^+3xy=26.  ^     (x^+3xy  = 

{Sy'^+2xy=  39.  '  \xy  +  4:y^  = 

^    (  4^2_^to/+4?/2=13.  .  ^    (  x^  +  xy  +  2  y^ 

'   I  8x'^-12xy+Sy^=ll.  [2x^+2x1/  +  y^ 


=  54. 
115. 


74. 
■12^?/+8/=ll.  "    12^24.2^^  +  ^=73. 

Queries.  What  is  a  homogeneous  equation  %  Into  what  forms 
may  simultaneous  quadratic  equations,  M'hich  can  be  solved,  be 
grouped  %  What  is  the  degree  of  the  equation  arising  from  eliminat- 
ing one  unknown  number  from  two  equations,  each  of  the  second 
degree  ?    Prove  it.     How  may  such  equations  be  solved  .^ 


SIMULTANEOUS   QUADRATIC  EQUATIONS.         351 

Note.  In  solving  the  following  equations  the  student  is  cautioned  not  to 
work  at  random,  but  to  study  the  equations  until  lie  sees  how  they  may  be 
combined  in  oixler  to  produce  sinjple  equations,  and  tlien  perform  the  opera- 
tions thus  suggested.  Usually  the  operations  of  addition,  subtraction,  multi- 
plication, division,  or  factoring  will  effect  a  simplification  of  the  equations. 


Miscellaneous  Exercise  129. 
Solve  the  following  equations : 


\xy=  15. 

11. 

^2^+:\u.^y-\-:\xy^-\-2y^  =  {). 
U2-r.t//  +  y2^1-x'^/. 

2   /^-3'  =  3- 

12. 

{.ly-\-  .I25x  =  y  —  x. 
I  y  —  .0  X  =  .7o  X y  —  o  X. 

^■\x-.>,=  l. 

13. 

f  .Sx  +  .l2by  =  ox-y. 
1  3  a:  -\-y  =  —2.25xy. 

<x>  +  y'  =  2U. 
^  \x  +  y  =  22. 

14. 

1  a:*  -f  7/4  =  706. 
\x  +  y  =  2. 

(a?  +  i/  =  7i. 
{  xy  =  o5. 

15. 

{x  +  y  +  x^-\-y^=\S. 
1  xy  =  6. 

^-   j(y_i)^.2_3^^2. 

16. 

{4(x  +  y)^3xy. 

1  X  +  y  -\-  x^-\-i/=  26. 

1  0^-7/2=  175. 

17. 

(a^+  ;/  ,,  337. 
\x-^y  =  7. 

(.,5+  5  7/2  =  6  a;.  ix^  +  xy-\-x=\4. 

\x^-5y^  =  4:xy.  ^^'  \  y^ -{- xy  +  y  =  2S. 

^    r  2:2  4.  ^y^  140.  ^^^  {2^-yS  =  20S. 

'  \y'^  +  xy  =  06.  "  I  xy(x  —  y)  =z48. 

10.  i^-^r!-  20.  (-?/  +  =^y  =  12- 


xy-y'  =  4.  ■  \y  +  sc>y=lS. 


352 
21. 


ELEMENTS  OF  ALGEBRA. 


22. 


^  -\-  if'  ^=^'^xy. 
X  +  y  =  b. 

2xy  +12  =  Zx\ 
6  xy  -{-  12  =  a;^ 

2       3 

23.  ^  ^      y 
lxy  =  2. 

x-\-  y 


,./: 


^4. 


24. 


25. 


26. 


27. 


29. 


30. 


31. 


32. 


l-\-  xy 

I  -  xy 

2^4 +  a;V  +  y/*  =  7371. 
x^  —  xy  -\-f  =  63. 

x^  +  y^  =  641. 
0^2/(^2  +  2/^)  =-  290. 

x^  -\-  3  a;  ?/  +  ;?/  =  19. 
a;2  +  ;^2  ^  lo; 

a;2  —  3  xy  +  ;/^.=  —  5. 
3^2_5^y+3y2^9 

y^  —  x^  =  a^. 
y  —  X  =  a. 

x^-\-y^  =  Ux^7/. 
X  +  y  =  a. 

96  —  x^y^  =  4:xy. 
X  +  y  =  6. 

x^  —  y/  =  56. 
x^^-  xy  \f-^  28, 


33. 

34. 

35. 
36. 

37. 
38. 
39. 
40. 
41. 
42. 
43. 
44. 


y'^  —  xy  =  a?  ■{■  W'. 
xy  —  x'^  =  2  ah, 

x^y  —  y  —  21. 
x^  y  —  X  y  —  6. 

^       2  x^      480 
2/2  +  "^  -  "49"  • 
(^a;2  +  2/^  =  65. 

1       1       a;  +  ?/ 
X       y  6 

a;  +  V  5 


/ 


x  -\-  y  -[-  I 


xP'y'^  +  5  a;?/  =  84. 
a;  +  ?/  =  8. 


■'«  +  y  +  Va:  +  y  =  12. 
x^  +  f^  41. 


a;  +  ?/  +  ^/x  +  y  =  12. 
x^  +  y'^  =  189. 

a;  +  Vxy  +  ?/  =:  19. 
x^  +  xy  +  y^  =  133. 

a/^  +  Vy  =  4. 
V^3  -^  ^f  =  28. 

a:?^  4-  2/5  =  6. 
a:l  +  7/1  =  126. 

?/2  —  a;2  =  4  a  &. 
xy  =  a^  —  h'^. 

2x?-  ?yxy  +  ?y2  =  4. 
2a;2/- 3  2/2-:i;2^-9. 


SIMULTANEOUS  QUADRATIC  EQUATIONS.  353 

48.  <i^      2^ 

La;       y 

X  -^  y      xj-_y  _  10 

49.  -{  x  —  y      X  +  y~   'S  ' 
y  +  /  =  45. 

X  •\-  y  ^  x  —  y 

50.  'ia-^b~a  —  b 
xy  =  ab. 


51.  The  sum  of  the  squares  of  the  digits  composing  a 
number  of  two  places  of  figures  is  25,  and  the  product  of 
the  digits  is  12.     Find  the  number. 

52.  There  are  two  numbers  whose  sum,  multiplied  by 
the  greater,  gives  144,  and  whose  difference,  multiplied  by 
the  less,  gives  14.     Find  the  numbers. 

53.  The  sum  of  the  squares  of  two  numbers  is  a,  and 
the  difference  of  their  squares  is  b.  Find  the  numbers. 
Solve  for  a  =  170  and  6  =  72. 

54.  A  number  divided  by  the  product  of  its  two  digits 
gives  the  quotient  2 ;  and  if  27  be  added  to  the  number, 
the  digits  are  reversed.     Find  the  number. 

55.  The  sum  of  two  numbers  is  a,  and  the  sum  of  their  4th 
powers  is  b.    Find  the  number.    Solve  for  a  =  4  and  b  =  82. 

56.  The  difference  of  two  numbers  is  a,  and  the  differ- 
ence of  their  cubes  is  7  a'     Find  the  numbers. 

57.  The  difference  of  two  numbers  is  3,  and  the  difiTer- 
ence  of  their  5th  powers  is  3093.     Find  the  numbers. 


354  ELEMENTS  OF  ALGEBRA. 

58.  A  number  consisting  of  two  digits  has  one  decimal 
place ;  the  difference  of  the  squares  of  the  digits  is  20,  and 
if  the  digits  be  reversed,  the  sum  of  the  two  numbers  is  11. 
Find  the  number. 

59.  Find  three  numbers  whose  sum  is  38,  such  that  the 
difference  of  the  first  and  second  shall  exceed  the  difference 
of  the  second  and  third  by  7,  and  the  sum  of  whose  squares 
is  634. 

60.  The  small  wheel  of  a  bicycle  makes  135  revolutions 
more  than  the  larger  wheel  in  a  distance  of  260  yards ;  if 
the  circumference  of  each  were  one  foot  more,  the  small 
wheel  would  make  27  revolutions  more  than  the  large 
wheel  in  a  distance  of  70  yards.  Find  the  number  of  feet 
in  the  circumference  of  each  wheel. 

61.  Find  two  numbers  such  that  their  difference  shall 
be  a,  and  the  product  of  their  n\k\.  roots  c.  Solve  for  a  =  4, 
c  =  2,  and  n  =  h. 

62.  Find  a  fraction  such  if  the  numerator  be  increased 
and  the  denominator  diminished  by  2,  the  result  will  be  its 
reciprocal;  while  if  the  numerator  be  diminished  and  the 
denominator  increased  by  2,  the  result  will  be  \%  less  than 
its  reciprocal. 

63.  A  principal  of  $10,000  amounts,  with  simple  in- 
terest, to  $14,200  after  a  certain  number  of  years.  Had 
the  rate  of  interest  been  1  %  higher  and  the  time  1  year 
longer,  it  would  have  amounted  to  $15,600.  Find  the 
time  and  rate. 

64.  A  sum  of  money  at  interest  amounted  at  the  end  of 
the  year  to  $10,920.  If  the  rate  of  interest  had  been  1  % 
less,  and  the  principal  $100  more,  the  amount  would  have 
been  the  same.     Find  the  principal  and  rate  of  interest. 


INDETERMINATE  EQUATIONS.  355 


CHAPTER   XXIV. 
INDETERMINATE  EQUATIONS. 

147.  Simple  Indeterminate  Equations  are  equations  of 
the  first  degree  that  admit  uf  aii  unlimited  number  of 
solutions. 

Thus,  in  3x-2y  =  2,  if  y  =  2,  x  =  2;  if  3/  =  3,  a;  =  2f  ;  if  y  =  5, 
X  =  4;  if  y  =  Si  X  =  6  ;  etc.  It  is  evident  that  an  unHmited  num- 
ber of  values  may  be  given  to  y  and  x  that  will  satisfy  the  equation. 
Hence,  an  ecj nation  containing  two  unknown  numbers  admits  of  as 
many  solutions  as  we  please,  and  is  indeterminate. 

Since  the  values  of  the  unknown  numbers  are  dependent  upon 
each  other,  they  may  be  confined  to  a  particular  limit ;  as,  for  exam- 
ple, suppose  the  variables  to  be  restricted  to  positive  or  negative  inte- 
gers, we  may  thus  limit  the  number  of  solutions. 

Example  1.    Solve  19x-i-by=  119,  in  positive  integers. 
Solution.     Transpose  19  a:,  5i/=119— 19  2:. 

Therefore,  y  =  23-3x  +  4l-^]     (1) 

l-x 
Since  the  value  of  y  is  to  be  integral,  then      -      must  be  integral, 

although  fractional  in  form;  and  so  also  is  any  multiple  of  it. 

Let  — r—  =  n,  an  integer. 

Therefore,  x=l-5n  (2) 

Substitute  in  (1),  y  =  20+19  n  (3) 

We  must  take  only  such  integral  values  for  n  as  will  give  positive 

integral  values  for  x  and  y. 

(2)  shows  that  n  may  be  0,  or  have  any  negative  integral  value, 

but  cannot  have  a  positive  integral  value. 


356  ELEMENTS   OF  ALGEBRA. 

(3)  shows  that  n  may  be  0  and  —  1,  but  cannot  have  a  negative 
integral  value  greater  than  1 . 

Therefore,  n  may  be  0  and  —  1. 

^^"^^'^=2i}'^^'%=i}' 

Query.     Can  n  be  —  2  or  +  1  ?     Why  ? 

Example  2.     Solve  7  x  —  l^y  =  19,  in  positive  integers. 

Process.     Transpose  and  solve  for  a;,     x  =  2-hy  +  bl  — ;=—  J    (I) 

Let  —j —  =  n,  an  integer. 

Therefore,  y  =  7n-l                   (2) 

Substitute  in  (1),  x=l2  7i-\-l                 (3) 

Evidently  x  and  y  will  both  be  positive  integers  if  n  have  any 
positive  integral  value. 

Hence,  x  =  13,  25,  37,  49,  ... . 

y=    6,  13,20,  27,  .... 

Notes :  1.  Having  obtained  a  few  of  the  possible  values  of  x  and  y,  the  law 
will  become  evident. 

2.  It  will  be  seen  from  the  above  solutions  that  when  only  positive  integral 
values  are  required,  the  number  of  solutions  will  be  limited  or  unlimited  ac- 
cording as  the  sign  connecting  the  terms  is  positive  or  negative. 

Example  3.     Solve  I90x  —  23y  =  708,  in  least  positive  integers. 

/3  -  x\ 
Process.     Solve  for  y,  y  =  Sx~30  —  6\—^j     (1) 

S  —  x 

Let  ^      =  n,  an  mteger. 

Therefore,  x  =  3-2Sn  (2) 

Substitute  in  (1),  y  =  -6-190  n  (3) 

Evidently  x  and  y  will  both  be  least  positive  integers  if  n  be  —  1 . 
Therefore,  n  =  —  ],  x  =  26,  and  y  =  184. 

Note  3.  If  the  coefficient  of  the  unknown  number  in  the  numerator  of  the 
fraction  is  not  1,  it  will  be  necessary  to  make  several  transformations. 

Example  4.    Solve  21a:+  17  y  =  2000,  in  positive  integers. 


»  ._       —  ",  ail  xiiiegei. 
3-n 

(2) 

3-»j 

— 7 —  =  w,  an  integer. 

n  =  3  -  4  m. 

x=  17m-10 

(3) 

3/=  130- 21  m 

(4) 

INDETERMINATE  EQUATIONS.  357 

11  -4x 
Solution.     Solve  for  y,  y  =  117  -  x  +  — r= —     (1) 

w       U-4x 
Transpose,  y  +  a:  — 117  =  — yj 

Since  x  and  y  are  to  be  integral,  y  +  x  —  111  will  be  integral ; 

11  —  4  a:      .,,  ,  , 

hence,  — r^ —  will  be  integral. 

ll-4a: 
Let 

Therefore, 

>.x       3-n  ,      . 

Now  —z —  mwtt  be  integral. 

Let 

Therefore, 

Substitnte  in  (2), 

Substitute  in  (1), 

(3)  shows  that  m  may  have  any  positive  integral  value,  but  can- 
not be  0,  or  have  any  negative  integral  value. 

(4)  shows  that  m  may  have  any  integral  value  from  0  to  6,  or  any 
negative  integral  value,  but  cannot  have  a  positive  integral  value 
greater  than  6. 

Therefore,  m  may  be  1,  2,  3,  4,  5,  Q,  giving  the  following  pairs  of 
values : 

x=      7,  24,  41,  58,  75,  92. 

y  =  109,  88,  67,  46,  25,     4.     Hence, 

To  Solve  a  Simple  Indeterminate  Equation,  Involving  Two 
Unknown  Numbers,  for  Integral  Values.  Find  tlie  value  of 
one  of  the  unknown  numbers.  Place  the  fractional  part  of  this  value 
efjual  to  Uj  an  integer,  and  solve  the  resulting  equation  for  the  other 
unknown  number.  Substitute  this  result  in  the  value  first  obtained. 
Solve  the  two  simple  equations  thus  fonuerl,  by  inspection,  for  inte- 
gral values  of  n. 

Notes:  4.  It  is  better,  in  solving  the  original  equation,  to  solve  for  the 
unknown  number  which  has  the  least  coefficient. 

5.  A  little  insrenuity  in  arranging  the  terms  will  often  obviate  the  necessity 
of  a  second  transformation. 


358  ELEMENTS  OF  ALGEBRA. 

148.  There  can  be  no  integral  values  of  x  and  y  in  an 
equation  of  the  form  ax  ±h  y  =  c,  ii  a  aiid  b  have  a  com- 
m6n  factor  not  common  also  to  c. 

For,  suppose  d  to,  be  any  factor  of  a  and  also  of  b,  but  not  of  c, 

sucli  that  a  —  md  and  h  =  n  d. 

c 
Then  mdx  ±.ndy  =  c,  or  mx  ztny  =  -j. 

Since  m  and  n  are  integers,  if  'x  and  y  be  also  integers,  mx  ±.  ny 

is  an  integer.      But  ^  is  a  fraction.      Hence,  no  integral  values  of 

X  and  y  can  be  found. 

Notes :  1.  If  a,  b,  and  c  have  a  common  factor,  it  should  he  removed  by 
division,  then  proceed  as  in  Art.  147. 

2.  The  solution  of  any  indeterminate  equation  of  the  form  ax  —  by  =.  ±c, 
in  which  a  and  h  are  prime  to  each  other,  is  always  possible,  and  admits  of  an 
unlimited  number  of  integral  solutions  (Ex.  2,  Art.  147).  If  the  equation  be 
of  the  form  ax-\-  hy  =  c,  the  number  of  results  Avill  always  be  limited ;  and, 
in  some  cases,  the  solution  is  impossible  (Ex.  1,  Art.  147). 

Exercise  130. 

Solve  in  positive  integers : 

1.  2x-\-3y  =  2o;    Ux  =  5y-7;    3x  =  8y-16, 

2.  5r?:+ ll2/  =  254;    9  a:  +  13  2/ =  2000. 

3.  15x-l7y=l;    13x-9y=:l;    9x-Uy=:10. 
Solve  in  least  positive  integers : 

4    3^  +  7y  =  39;    3x+4y=:39;    7x+lDy  =  22^. 

5.  27a;-192/  =  43;    2^  +  7?/ =125;  555y-22a:  =  73. 

6.  19  ^-5?/ =119;    I7x  =  4:9y-S. 

Are  integral  solutions  possible  for  the  following?  Why? 

7.  3^  + 21 7/ =  1000;    7  a; +14?/ =  71. 

8.  323  a?- 527  y=  1000;    166  a:  -  192  y  =  91. 


INDETERMINATE  EQUATIONS.  369 

9.  Solve  7  a-  4-  15  y  =  145,  in  positive  integers,  so  that 
X  may  be  a  multiple  ot  y. 

^  146  ,  145 n 

Suggestion.    Let  x-ny,  then  y  =  ^      ^^,  and  x  =  ^^^^^. 

10.  Solve  'S9  X  —  6  y  =  12,  in  positive  integers,  so  that 
y  may  be  a  multiple  of  z, 

11.  Solve    20  a:  —  31 2/  =  7,  so  that  x  and  y  may  be 
positive,  and  their  sum  an  integer. 

Suggestion.     Put  x  +  y  =  n. 

149.    A  problem  is  indeterminate  when  it  involves  less 
conditions  than  there  are  unknown  numbers. 

Exercise  131. 

1.  Find  a  number  which  being  divided  by  3,  4,  and  5, 
gives  the  remainders  2,  3,  and  4,  respectively. 

Solution.     Let  x  represent  the  number  and  y  the  sum  of  the 
quotients,  then, 

x-2      a--3      x-4 
3     +^  +  — =  »• 

f'S-\-y\ 
Simplify  and  solve  for  ar,  a-  =  y  +  2  4- 13  I  — j;^  I    (1) 

3-l-w  \  •*'   / 

Let  -^=-  =  n,  an  integer. 

Therefore,  y  =  47  n  —  3. 

Substitute  in  (1).  a:  =  60  n  -  1. 

Hence,  n  may  be  1,  2,  3,  4,  etc. 

Therefore,  x  =  59,  119,  179,  239,  etc. 

.7  =  44,    91,  138,  185,  etc. 

2.  Find  the  least  number  which  being  divided  by  2,  3, 
4,  5,  and  G,  gives  remaindei-s  1,  2,  3,  4,  and  5,  respectively. 

3.  Find  two  numbers  which,  multiplied  respectively  by 
14  and  18,  have  for  the  sum  of  their  products  200. 


360  ELEMENTS  OF  ALGEBRA. 

4.  Divide  142  into  two  parts,  one  of  which  is  divisible 
by  9,  and  the  other  by  14. 

5.  There  are  two  unequal  rods,  one  5  feet  long  and  the 
other  7.  How  many  of  each  can  be  taken  to  make  up  a 
length  of  123  feet  ? 

6.  Find  two  fractions  having  5  and  7  for  denominators, 
and  whose  sum  is  ||. 

7.  Find  the  least  number  that  when  divided  by  9  and 
17  will  give  remainders  5  and  12,  respectively. 

N-  5 
Suggestion.     Let  N  represent    the    number,  — - —  =  x,  and 

N~  12 

— p^ —  =  y.     .'.  9x=l7y-{-1. 

8.  A  farmer  bought  sheep,  pigs,  and  hens.  The  whole 
number  bought  is  125,  and  the  whole  price,  $225.  The 
sheep  cost  $5,  the  pigs  $2.50,  and  the  hens  25  cents. 
How  many  of  each  did  he  buy  ? 


Solution.     Let 

X  =  the  number  of  sheep, 
y  =  the  number  of  pigs, 

and 

z  =  the  number  of  hens. 

Then, 

x  + 

y  +  z=  126                           (1) 

and 

5x+2. 

■5y  + 

.25  z  =  225                             (2) 

From  (1)  and  (2), 

?/=86-2x-^       (3) 

Let 

x-1 
Q     -  n,  an  integer. 

Therefore, 

x  =  9n-\-l. 

Substitute  in  (3), 

y  =  M-l9n. 

Substitute  in  (1), 

2;=  40+  10  n. 

Therefore,  n  may  be  0,  1,  2,  3,  and  4,  giving  the  following  values : 

x=  I,  10,  19,28,37. 
y  =  84,  65,  46,  27,  8. 
z  =  40,  50,  60,  70,  80. 


PROBLEMS.  361 

Qaeries.  How  many  solutions  ?  In  how  many  diflFerent  ways 
may  the  stock  l)e  bought?  How  solve  by  means  of  only  two  un- 
known numbers? 

9.  How  can  one  pay  a  sum  of  $  1.50  with  3  and  5  cent 
pieces  ?     In  how  many  ways  can  the  sum  be  paid  ? 

10.  Can  a  grocer  put  up  the  worth  of  S3.50  in  11  and 
7  cent  sugar  ?  In  how  many  ways  can  he  do  it  in  even 
and  odd  pounds,  respectively  ?  Find  the  greatest  and  least 
number  of  pounds  of  the  7- cent  sugar  he  can  use. 

11.  Is  it  possible  to  pay  £50  by  means  of  guineas  and 
three-shilling  pieces  only  ? 

12.  A  owes  B  $5.15.  A  has  only  50-cent  pieces  and  B 
only  3-cent  pieces.     How  may  they  settle  the  account  ? 

13.  A  farmer  bought  horses  at  S  60  a  head  and  sheep  at 
S8,  and  found  that  he  bad  invested  S4  more  in  sheep  than 
horses.     How  many  of  each  kind  did  be  buy  ? 

14.  A  farmer  invested  $1000  in  75  head  of  cattle,  worth 
$25,  $15,  and  $10  per  head.  Find  the  number  of  each 
kind,  and  the  number  of  ways  in  wliich  he  could  buy 
them. 

15.  A  grocer  had  an  order  for  75  pounds  of  t€a  at  55 
cents  a  pound,  but  having  none  at  that  price  he  mixed 
some  at  30  cents,  some  at  45  cents,  and  some  at  80 
cents.  How  much  of  each  kind  did  he  use,  and  in  how 
many  ways  can  he  mix  it? 

16.  How  many  pounds  of  20,  35,  and  40  cent  coffee 
must  a  grocer  take  to  make  a  mixture  of  150  pounds  worth 
30  cents  a  pound  ?  In  how  many  ways  can  the  mixture 
be  made  ? 


362  ELEMENTS  OF  ALGEBRA. 

17.  How  many  gallons  of  S  1.50,  S1.90,  and  $1.20  wine 
must  a  vintner  take  to  make  a  mixture  of  40  gallons  worth 
$1.60  per  gallon  ?  How  many  ways  may  the  mixture  be 
made  ?  Can  an  odd  number  of  gallons  of  each  kind  be 
taken  ?     An  even  number  ? 

18.  In  how  many  ways  can  £1  be  paid  in  half-crowns, 
shillings,  and  sixpence,  the  number  of  coins  in  each  pay- 
ment being  18  ? 

19.  A  hardware  merchant  paid  $180  for  20  stoves. 
There  were  three  sizes:  one  $19  each,  another  $7,  the 
other  $6.     How  many  of  each  size  did  he  buy? 

20.  A  person  having  a  basket  of  oranges,  containing 
between  50  and  72,  takes  them  out  4  at  a  time,  and  finds  1 
over;  he  then  takes  them  out  3  at  a  time,  and  finds  none 
over.     How  many  had  he  ? 

A^-l  N 

Suggestion.  Let  N  represent  the  number,  — j—  =  x,  and  -«  =  t/. 

l+x      ^      I  +  x 
.'.  y  =  x+  ~^'     Pwt  — ^  =  n.     Then  n  must  be  5  or  6. 

21.  A  poultry  dealer  has  a  basket  containing  between 
200  and  300  eggs,  he  finds  that  when  he  sells  them  13  at  a 
time  there  are  9  over,  but  when  he  sells  them  17  at  a  time 
there  are  14  over.     Find  the  number  of  egfjs. 

22.  Two  countrymen  together  have  100  eggs.  If  the 
first  counts  his  by  eights  and  the  second  his  by  tens,  there 
is  a  surplus  of  7  in  each  case.     How  many  eggs  has  each  ? 

23.  A  surveyor  has  three  ranging  poles  of  lengths  7  feet, 
10  feet,  and  12  feet.  How  may  he  take  40  of  tliem  to 
measure  113  yards?  In  how  many  ways  may  the  mea- 
surement be  made  ? 


INEQUALITIES.  363 

CHAPTER   XXV. 
INEQUALITIES. 

150.  Since  a  positive  number  is  greater  than  any  negative  num- 
l)er,  the  statement  that  a  is  algebraically  greater  than  6,  or  that  a—h 
is  positive,  is  expressed  by  a  >  6  ;  that  a  is  algebraically  less  than  6, 
or  that  a  —  6  is  negative,  is  expressed  by  a  <  6.     Hence, 

An  Inequality  is  a  statement  that  on^  expression  is 
greater  or  less  than  another;  as, 

1  —  ar  > = — :    m  —  n  <  x. 

The  expression  at  the  left  of  the  sign  is  calle<l  the  first  member, 
and  the  expression  at  the  right,  the  second  member  of  the  in- 
equality. 

The  fonn  a>  h>  c,  means  that  6  is  less  than  a  but  greater 
than  c. 

Notes  :  1.  Inequalities  are  said  to  subsist  in  the  same  sense  wlien  the  lirst 
member  is  the  greater  in  each,  or  the  first  member  is  the  less  in  each  ;  as, 
3  >  2,  7  >  5,  and  .5  >  3 ;  a<h,  c<d,  and  m<n. 

2.  Two  inequalities  are  said  to  subsist  in  a  contrary  sense  when  the  first 
member  is  the  greater  in  one,  and  the  less  in  the  other  ;  as,  5  >  3  and  o  <  6  ; 
m  <  5  and  h  >  n. 

3.  An  inequality  is  said  to  be  solved  when  the  limit  to  the  value  of  the 
unknown  numl)er  is  found. 

151.  Subtract  a  +  &  from  each  member  of  a  >  6, 

then,  a  -  (a  +  b)  >  b-  (a  -\-  6). 

Simplify,  .  —  6  >  —  a, 

or,  ~  a  <  —  b      Hence, 

I.  If  each  member  of  an  inequality  has  its  sign  changed,  the  sign  of 
inequality  will  be  reversed. 


364  ELEMENTS  OF  ALGEBRA. 


Multiply  each  member  of 

-  5  <  5  by  - 

-2, 

then, 

10  >  -  10. 

Multiply  each  member  of 

a  >  6  by  - 

-m, 

then, 

—  am  <  —  6m. 

Divide  each  member  of 

-  6  <  4  by  - 

-2, 

then, 

3  >  -  2. 

Divide  each  member  of 

a  >  ft  by  - 

-m, 

then. 

a            b 

m           m' 

He 

II.    If  each  member  of  an  inequality  be  multiplied  or  divided  by 
the  same  negative  number^  the  inequality  tvill  be  reversed. 

Suppose  a>  b,  c  >  d,  m'>  n,  — 

By  definition,  a  —  b,  c-d,  m-n,  are  positive. 

Add,  (^a  -b)  +  {c  -  d)  +  (m  -  n)  +  ....  is  positive. 

or,  (a  +  c  +  m  -I .)  -  (b  +  d  -\-  n  -\-  ... .)  is  positive. 

Therefore  (by  definition),  a  +  c  -^  m  +  —  >  b  +  d  +  n  +  .,», 
Thus,  7  >  3 

5>  2 

4>  1 


Add,  16  >  6,  or  divide  by  2,  8  >  3.     Hence, 

III.  If  the  corresponding  members  of  several  inequalities  be  added, 
the  sum  of  the  greater  members  will  exceed  the  sum  of  the  lesser 
members. 

Suppose  a  >  b  and  m  >  n,  then  a  -  b  and  m  —  n  are  positive. 
But,   (a  —  b)  —  (m  —  n),    or   (a  —  m)  —  (6  —  n)    may   be   either 
positive,  negative,  or  0. 

Therefore,  a  —  m  >  b  —  n,  a  —  m  <  b  —  n,  or  a  —  m  =  b  —  n. 


Thus,  5  >  3 

3>  2 


Subtract,    2  >  1 


7>4 
5>  1 

Subtract,    2  <  3 


8>  7 
6>5 

Subtract,    2  =  2.    Hence, 


IV.  If  the  members  of  one  inequality  be  subtracted  from  the  corres- 
ponding members  of  another,  tht  resulting  inequality  will  not  always 
subsist  in  the  same  sense. 


INEQUALITIES.  865 

1  2  X       3  z      64 

Example  1.     Solve  3^  x ^ —  >  -^^  +  ^^  for  the  limits  of  x. 

Solution.    Free  from  fractions  and  simplify, 

112a;-6>45x+  128. 
Subtract  45  x  -  6,  67  a;  >  134. 

Divide  by  67,  x  >  2. 

Therefore,  x  is  greater  than  2. 

Example.  2.     Solve  the  following : 

\bx-Zy>Zx-^b   (1) 

i3a:  +  t/=r22  (2) 

Solution.     Subtract  3  x  from  (1),  2  x  -  3  y  >  5  (3) 

Multiply  (2)  by  3,  9x  +  3  y  =  66  (4) 

Add  (3)  and  (4),  11  x  >  71. 

Divide  by  11,  x  >  Q^^. 

22 -w 
From  (4),  x  =       ^      • 

Substitute  in  (3)  and  simplify,  —y>-\\. 

Therefore,  2/  <  2^^  (see  I) 

Example  3.     Solve  the  inequalities : 

{fix  —  mn^n^  —  mx     (1) 
\  mx  ~  nx  +  mn  <  in^     (2) 

Process.     Simplify  (1)  and  solve,  x  >  n. 

Simplify  (2)  and  solve,  x  <rr. 

Hence,  x  is  greater  than  n  and  less  than  m. 

Note.  The  principles  applied  to  the  solutions  of  equations  may  be  applied 
to  inequalities,  except  that  if  each  member  of  an  inequality  haa  iti  sign 
changed,  the  lign  of  inequality  will  be  reversed. 


Exercise  132. 
Find  the  limit  of  x  in  the  following : 
1.   4x-3  >fa;-f;    f  —  |  ^  <  9  -  3  a:. 
^o  o       o  1      x^  —  a      a  —  X       2x      a 


366  ELEMENTS  OF  ALGEBRA. 

o  ax  —  2b       a  X  —  a       ax       2 
^-  Zh  2h~  ^  1"  ~  3  * 

4.  If  2^2  +  4  a;  >  12,  show  that  x>2. 

5.  If  7  a;2  -  3  a;  <  160,  show  that  x  <  o. 

6.  If  4  ic  +  12  —  ic^  >  0,  show  that  x  is  included  be- 
tween 6  and  —  2. 

7.  If  9  a;  <  20  it^  +  1,  show  that  x  >  \  oi  <  \. 

8.  If  15  —  ic  —  2  ic^  >  0,  show  that  x  lies  between  | 
and  —  3. 


Q    I  ^  ;^  >  30  —  4  ic.  ^Q    (  1 J  ^  <  I  ^  +  3|-. 

<  3  :?:  +  49.  *   (  6  oj  >  24  -  2  ^. 


Find  an  integral  value  of  x  in  the  following 

\  IQx 

(  |(^  +  2)  +  ^  <  I  (^  -  4)  +  9. 
'   \l(x-\-2)  +  \x>l{x+l)^^. 

Find  the  limits  of  x  and  y  in  the  following : 

.o    i3^4-5^>121.  |7.^;  +  5^>19. 

*   (4a;+  72/ =168.  /  a;  -  3/ =  1. 

(«  +  &)  ^  —  (a  —  ?))  2/  >  4  a  &. 


^       I  (tt  -  &)  ^  +  (a  4- 


h)y  =  2(a  +  h)  {a  -  6). 


15.  A  certain  number  plus  5,  is  greater  than  one  third 
the  number  plus  55  ;  while  its  half  plus  2,  is  less  than  41. 
Find  the  number. 


INEQUALITIES.  367 

16.  Find  the  price  of  oranges  per  dozen,  when  three 
times  the  price  of  one  orange,  decreased  by  tiiree  cents,  is 
more  than  twice  its  price  increased  by  one  cent ;  and  eight 
times  the  price  of  one  orange,  decreased  by  twenty  cents; 
is  less  than  three  times  its  price  increased  by  ten  cents. 

152.  Since  the  stjuare  of  a  negative  number  is  positive,  if  a  and  b 
represent  any  two  numbers,  (a  —  6)^  must  be  positive,  whatever  the 
values  of  a,  and  b.  Therefore,  since  every  positive  number  is  greater 
than  zero, 

(a  -  6)2  >  0. 

Expand,  a^  -  ^ah +  h'^>  0. 

Add  2  a  6  to  each  member,  a*  +  6^  >  2  a  6.     Hence, 

The  sum  of  the  squares  of  two  unequal  numbers  is  greater 
than  tvnce  their  product. 

Vote.    The  above  is  a  fundamental  principle  in  inequalities. 

Example  1.  Show  that  a"^  -\- 1)^  ■\-  c^  >  ab  +  a  c  +  h  c,  a  and  h 
positive. 

Prool     Since  a,  b,  and  c  are  any  unequal  numbers, 

a2+62>2a6  (1) 

a«+c«>2ac  (2) 

62  +  c«>26c  (3) 

Add  the  corresponding  members  of  (1),  (2),  and  (3), 

2aa  +  2  62-l-2c2>2a6+2ac  +  2  6c. 

Divide  by  2,  a«  +  6«  +  c^  >  a  6  +  a c  +  6c. 

Query.     How  if  a  =  6  =  c  ? 

Example  2.     Show  that  a«  +  6«  >  a*  6  -}-  a  6*. 

Proof.     We  shall  have,  a*  +  }fi  >  a'^b  +  a 6*. 

Factor,  (a  +  6)  (a«  -  a  6  +  6^)  >  a  6  (a  +  6). 

Divide  by  a  +  6,  a'^  —  ah -\- b"^  >  ab. 

Add  a  6,  a«  +  6^  >  2  a  6. 

Therefore,  a«  +  6»  >  a^  6  +  a  6*. 


368  ELEMENTS   OF  ALGEBRA. 

Example  3.     Which  is  the  greater,  V/ 1-  1/  — r-  or  /y/a i 

Proof.     We  shall  have, 

Square  each  member, 
a2  62 


2  Y a bmn  -\ r-  >  or  <  a 6  +  2 \/a bmn  +  m n. 

it9f    tV  CI    O 

, a^h^      m^n^ 

Subtract  2  \/ahmn^ 1-  — r-  >  or  <  ao  +  ww. 

^  m  n         a  0 

Free  from  fractions  and  factor, 

{ah  +  m  n)  {a^ h^  —  abmn  -\-  m^ n^)  >  or  <  abmn(ah +  mn). 

Divide  by  ab  -\-  mn, 

a^h^  —  abmn  +  m^n^  >  or  <  ab  mn. 

Add  a  b  7«  n,  a^  b^  +  m^  n^  >  or  <  2  a  6  m  n. 

But,  a^b^-i  m'^n^':>  2abmn. 

Therefore,  \  1^  +  \  -afT  >  ^""^  -^  V^""- 


Exercise  133. 

Show   that,   the   letters    being   unequal,   positive,    and 
integral : 

h^       a^       a       b 

2.  a  6-2  +  a-%  >  a-^+  b-^ ;  (ni^-]-  n^)  (mH  n^)  >  (m^+  n^f. 

3.  xy-\-xz-\-yz  <  {x  +  y^z)^-\-{x+z—yf'-{-{y  +  z~x)\ 
Which  is  the  greater : 

.       „     .  „  ,         a  +  b        2  ab    m        n  11 

4.  71^4-1  or  ?i2+  ^ ;  —^  or  —-7 ;  -5 s  or 

5.  1±J  or  ^^;    3(1  +  a2  +  a^)  or  (1  +  c^  +  a'f, 

ga^  y       x^  —  y^  ^ 


INEQUALITIES.  369 

1/9         a/^ 
V3       v5 
Queries.     How  in  4  and  6,  if  a  =  6  ?    In  4,  if  n  =  1  ? 

7.  If  a^  +  ^>2  +  c2  =  1,  and  a?»  +  y»  +  «2  =  1,  show  that 
aa;  +  &y  +  C2  <  1. 

Query.     How  ifa  =  6  =  c  =  a:  =  y  =  z? 

If  a  >  6,  show  that : 

8.  a  -  6  >  iVa  -  V6)^  a^  +  7  aH  >  (rr  +  &)» 

9.  a-6*  >  a*6« ;    a^  +  13  a  ^2  >  5  a26  +  U  63. 

Miscellaneous  Exercise  134. 
Example  1.    Solve  the  inequahties : 


{: 


V2(xy-f-y2)  +  4<;y(2i/-l)(y  +  a:)    (1) 
2x  +  5y>8  (2) 

Solution.     Square  each  member  of  (I)  and  simplify, 
2xy-\-2y^-\-  4  <2y^-^2xy  -y-x. 
Subtract  2ari/  +  2  y«,  4  <-y-x  (3) 

Multiply  each  member  of  (3)  by  2, 

8<-2y-2a:  (4) 

Add  the  corresponding  members  of  (2)  and  (4), 

3y  >  16.     .-.  y  >5f 
Multiply  each  member  of  (3)  by  5, 

20<-5y-5a:  (5) 

Add  (2)  and  (5),  -  3a:  >  28,  or  3a:  <  -28.    .-.  x  <  -9f 

Example  2.     Simplify  (y-\-x<m  —  n)  (m*  +  m  n  4- 71^  >  y  —  x). 

Solution.    We   are    to    multiply   the    corresponding    members 
together,  (y  +  x)  (y  —  x)  =  if  —  a:*, 

(m  —  n)  (m*  ■\-  mn  -\-  n'-*)  =  m*  —  n*. 
Therefore, (y +x < m-n){mHmn-\- n^ >y-x)  =  y^~x^  <  m*- n». 

24 


370  ELEMENTS  OF  ALGEBRA. 

Example  3.    Which  is  the  greater,  x^  +  y^  or  x^y  -\-  y*xl 

Proof.     We  shall  have,  x^  +  y^>  or  <:  x*  y  -{-  y*  x. 

Subtract  x^y  -\-  y^x,  x^  —  x^  y  +  y^  —  y*  x  >  or  <  0. 

Factor,  (x^  -  y^)  (x-y)>  or  <  0. 

Now,  whether  a;  >  or  <  .y,  the  two  factors,  x^  —  y^  and  x  —  y,  will 
have  the  same  sign.     Hence,  since  (x'^—y^)  i^~y)  is  always  positive, 

{x^  -  y^)  {x-y)>  0. 
Therefore,  a;^  +  2/^  >  x^  y  -\-  y^x. 

Example  4.     Which  is  the  greater,  m*  —  n*  or  4  n^  (^  _  ,^)  when 
m  >  n  ? 

Proof.     We  shall  have,  m^  —  n*  >  or  <  Am\m—n). 

Divide  by  w  —  n,  m^  +  wi*n  +  m  w^  +  n^  >  or  <  4  m^. 

Subtract  wi^  +  m^  n  and  factor  the  resulting  inequality, 

n^  (m  +  n)  >  or  <  m\^m—n). 
But,  m  >  w  (1) 

Square  (1),  m^  >  n^  (2) 

Multiply  (1)  by  2,  2  m  >  2  n. 

Add  m  —  n,  3m  —  ??>m  +  n  (3) 

Multiply  the  corresponding  members  of  (2)  and  (3), 

m^  (3  m  —  n)  >  n'  (m  -{-  n). 
Therefore,  4m^  (m  —  n)  >  w*  —  n*. 

6.    Find  the  sura  of  x^  +  y  >  1  —  a,  y^  —  2  a  >  5  +  4, 
^  X  +  y  <  2  a  +  1,  and  y^  -  S  x^  <  5  -  a. 

6.  From  a^  +  2  a  a;^  <  5  take  a  (a  +  x^)  y  n^  —1. 

7.  From  a2  <  3  -  .7.2  subtract  2  ^2  >  5. 

Multiply : 

8.  {a  +  hf  >  {x-yf  by  -3;    Z-f  <  5-^^  by  x^  +  y\ 

9.  Divide  a^-l^  >  a^+  h^  by  a2  +  ^2 
10.   Divide  11  a2  +  88&  >  121^2  ^y  -  H. 


INEQUALITIES.  371 

Perform  the  indicated  operations  and  simplify : 

11.   (w-l<  5)(m  +  l<10);    (a  <  n +  h){n-b  >  c). 


12.  (_  2  >  -  3)3 ;    (5  >  2)  -f-  (3  <  4)  ;    V25  >  9. 

13.  [-  243  >  -  1024]i  ;    (71  +  1)^  >  n^  -  n^  +  4:  n. 

14.  m3  -  7i3  >  (m  -  n)  {m^  +  n^) ;    4^-  64  <  8. 

15.  (m2  -  n^  <  u.^)'T-{ix>  m  +  n);    [-  ?t   >  i/f. 

Solve : 

16.  {X''2f  >  0^+  6 x-25',V(x-l)^  +  'S 2^  +  6  >2xK 

17.  a;  -  2  >  V  ^-^=^ ;    V3  -  4  Vi  >  VI6  2:  -  5. 


^g    |3y+2a;>3.         ^^    |  a:  +  f  >  Vo^  -  3  a;  +  y. 
(  4  >  4  7/  +  ic.  ■    (  5  >  a:  —  y. 

j42/-a^>?/  +  4.  J3:r-l>a:  +  3y. 

^"-    (3a;-6y>  l-4i/.       "^       (  27/-3rr2  =  3a;-3.x2. 

22.  38a:-7-15a:2<0;    6  ar^  4- 7  ;r  +  2  <  0. 

23.  17  a;- 6r»-5<0;    6  a:  +  11  -  a:^  <  2  a;  -  10. 
Find  integral  values  of  x  in  the  following : 

(3ix-.5x>5.  {x  +  7 

^^-    l2.5a:  +  ia;<8.         ^^'    (2a;  + 


a;  +  7  <  15. 

10  >  20. 


26    U^-i^<3.  27    |2a:-5>31. 

I  7a;-15>4a;+30.    ^'    (3a;-20<2ar. 

28.   ar2  4- 2a;-15<0;    a:^  ^  lOa:  4.  63  <  0. 


372  ELEMENTS  OF  ALGEBRA. 

Show  that: 

29.    Vl9+V3>  VIO  +  VT;  V5  +  Vn  >  V3+3V2c 

li  a  >h,  show  that : 


30.  Va^  -  62  +  Va^  -  (a  -  hf  >  a. 

31.  a^-h^  <3a^{a-  h)  and  >Sb^{a-  b). 

32.  a-h>  -j-^  and   <  -^3-  . 

If  x^  —  a^  +  y^,  if'  —  (p-  -^  d'^^  show  that: 

33.  xy'^ac  +  bdoTad  +  bc. 
Show  that: 

34:.   {a  b  +  X  I/)  (a  X  +  b  y)  ^  4:  a  b xy, 

35.  (ti  +  6)  (ft  +  c)  (6  +  c)  >  8  a  &  c. 

36.  Show  that  the  sum  of  any  fraction  and  its  reciprocal 
is  greater  than  2. 

?7.  In  how  many  ways  may  a  street  20  yards  long  and 
15  wide  be  paved  with  two  kinds  of  stones ;  one  kind 
being  3f  feet  long  and  3  wide,  the  other  4|  feet  long  and 
4  wide  ? 

38.  A  and  B  set  out  at  the  same  time  to  meet  each 
other;  on  meeting  it  appeared  that  A  had  travelled  a  miles 
more  than  B,  and  that  A  could  have  gone  B's  distance 
in  n  hours,  and  B  could  have  gone  A's  distance  in  m 
hours.  Find  the  distance  between  the  two  places.  Solve 
when  ft  =  18,  ^  =  378,  and  m  =  672. 


SERIES.  373 

CHAPTER   XXVI. 
SERIES. 

153.  A  Series  is  an  expression  in  which  the  successive 
terms  are  formed  according  to  some  fixed  law ; 

As,  1,  2,  4,  8, ,  in  which  each  term  is  double  the  preceding 

term  ;  a,  a  +  c/,  a  +  2d,  a  +  3d,  ,  in  which  each  term  exceeds 

the  preceding  term  by  d. 

ARITHMETICAL  PROGRESSION. 

154.  The  expressions  1,  6,  9,  13,  17, ,  and  16,  10,  6,  0, 

—  5,  —10,  ....,  are  called  arithmetical  progressions  or  series.  The 
first  is  an  increasing  series,  and  the  second  a  decreasing  series. 
The  general  form  for  such  a  series  is, 

a,  a  +  d,  a-\-^d,  a  +  3<i,  a  +  4ci,  a  +  Hd,  a  +  6cl,  .... 

in  which  a  is  the  first  term  and  d  the  common  difference ;  the  series 
will  be  increasing  or  decreasing  according  as  d  is  positive  or  negative. 
Hence, 

An  Arithmetical  Progression  is  a  series  in  which  the  adja- 
cent terms  increase  or  decrease  by  a  common  difference. 

In  every  arithmetical  series  the  following  elements  occur,  any 
three  of  which  being  given,  the  other  two  may  be  found  : 

The  first  term,  or  a. 

The  last  term,  or  /. 

The  common  difference,  or  d. 

The  number  of  terms,  or  n. 

The  sum  of  the  terms,  or  s. 


374  ELEMENTS  OF  ALGEBRA. 

By  an  examination  of  the  general  form  it  is  seen  that  the  coefficient 
of  d  is  always  1  less  than  the  number  of  the  term. 
Thus,  the  2d  term  is  «  +  rf,       or  a  +  (2  —  1)  cf, 
3d  term  is  a  +  2  </,    or  a  +  (3  —  1)  rf, 
4th  term  is  a  f-  3  </,    or  a  +  (4  —  1)  6?, 
12th  term  is  a  +  11  rf,  or  a  +  (12  —  1)  d,  and  so  on. 
In  the  nth,  or  last  term,  the  coefficient  of  c?  is  n  —  1.     Hence, 

To  Find  the  Last  Term  of  an  Arithmetical  Series,  when  the 

first  term,  the  common  difference,  and  the  number  of  terms 

are  given. 

I  =  a+  {n-l)d  (!) 

Note.     The  common  difference  may  always  be  found  by  subtracting  any 
term  of  the  series  from  that  which  immediately  follows  it. 

Example  1.     Find  the  18th  term  of  the  series  |,  f,  |,  etc. 

Process.     Here,  n  =  18,  a  =  |,  and  d  —  ^  —  l  =  \. 
Substitute  these  values  in  (i),  /  =  |  +  (18  -  1)  J  =  4. 

Example  2.    Find  the  30th  term  of  the  series  x  -\-  y,  x,  x  —  y,  etc. 

Process.     Here,  n  =  30,  </  =  x  —  {x+y)  =  —y,  and  a  =  x  +  y. 
Substitute  these  values  in  (i),  I  =  x  +  y  +  (30  —  l)(—y)  =  x  —  28y. 


Exercise  135. 

Find: 

1.  The  15th  term  of  7,  3,  - 1, .... 

2.  The  27th  and  41st  terms  of  5,  11,  17,  .... 

3.  The  20th  and  13th  terms  of  -  3,  -  2,  -  1,  .... 

4.  The  37th  and  89th  terms  of-  2.8,  0,  2.8, .... 

5.  The  40th  term  of  2  a  —  h,  4  a  -  3h,  6  a  —  5h,  .. 

6.  The  15th  and  8th  terms  of  J,  1 .  | 


ARITHMETICAL  PROGRESSION.  375 

7.  The  first  term  is  J,  the  102d  is  18.  Find  the  com- 
mon difference. 

8.  The  21st  term  is  53,  and  the  common  difference  is 
—  2J.     Find  the  first  term. 

9.  The  first  term  is  5^,  and  the  common  difference  is  3 J. 
What  term  will  be  42  ? 

10.  The  first  term  is  ^,  the  common  difference  is  ||,  and 
the  last  term  is  17^.     Find  the  number  of  terms. 

11.  The  54th  and  4th  terms  are  -  125  and  0.  Find 
the  42d  term. 

12.  Find  three  terms  whose  common  difference  is  J, 
such  that  the  product  of  the  second  and  third  exceeds  that 
of  the  first  and  second  by.  1  J. 

155.    Taking  the  elements  as  given  in  Art.  154 : 

s  =  a  +  (a  +  rf)+(«  +  2</)  +  (a  +  3rf)+(a  +  4f/)  +  ••..  ^ 
or  5  =  /  +  (;-r/)-f  (/-2c?H(/-3£/)  +  (/-4cO  +  .-..  a- 

Add,  2s  =  {a  +  [)-\-{a  +  l)-\-{a-\-l)  +  (a  +  l)  +  (a  +  l)  -f  ....  to  n  terms, 
or  25  =  n(a  +  /)  (1) 

Substitute  the  value  of  I  from  (i)  (Art  164)  in  (I), 
a  »  =  n  [2  a  +  (n  —  1)  d].     Hence  (solve  for  s), 

To  Find  the  Sum  of  all  the  Terms  of  an  Arithmetical  Seriei 

»  =  ^[aa-f  (n-l)<f]  (iH) 

Example  1.     Find  the  sum  of  an  arithmetical  series  of  17  terms, 
the  first  term  being  5^,  and  the  last  term  25 i. 
Process.     Here,  n  =  17,  a  =  5^,  and  /  =  2.''>|. 
Substitute  these  values  in  (ii),  «  =  y.  (5^  +  25^)  =  263j. 


376  ELEMENTS   OF  ALGEBRA. 

Example  2.     Find  the  sum  of  the  series  3|,  1,-1^, ,  to  19 

terms. 

Process.     Here,  n  =  19,  a  =  3^,  and  c?  =  1  —  3^  =  —  2^. 
Substitute  these  values  in  (iii), 

s=  ^-[2  X  3^  +  (19  -  1)(-  2^)]  =  -  361. 

12                3 
Example  3.    Find  the  sum  of  m ,  3  m ,  5  m ,  . . . . ,  to 

m'  m  m'        ' 

m  terms. 

Process.     Here',  n  =  m,  a  =  m  —  —  ,  and  d  =3m  —~  ~    m  —  -) 

=  2m  —  -  • 
m 

Substitute  in  (iii), 

2  m^  —  m  —  1 


=fK'"-9+^'" -'>("" -9] 


Example  4.  The  first  term  of  a  series  is  3  m,  the  last  —  35  m,  and 
the  sum  — 320  7W.  Find  the  number  of  terms  and  the  common 
di  Herence. 

Process.     Here,  s  =i  —  320  m,  a  =  3  m,  and  /  =  —  35  m. 
Substitute  in  (ii), 

-  320 m  =  ■x(S  m  -  35  m)  =  -  16  mn.  .-.  n  =  20. 
Substitute  in  (iii), 

-  320  m  =  %0-  [6  m  +  19  rf]  =  60  m  +  190  c?.     .-.  d  =  -2m. 

Example  5.     How  many  terms  of  the  series  —  C|,  —  6f ,  —6, , 

must  be  taken  to  make  -  52|  ? 

Process.     Here,  s=  —  52|,  a  =  —  6|,  and  c/  =  f . 

n 
Substitute  in  (iii),  -  52|  =  ^  C"  ¥  +  (n  -  1)  X  f  ]. 

Simplify  and  solve  for  n ,  n  =  1 1  or  24. 

Query.  Do  both  of  these  values  satisfy  the  conditions  ?  In 
explanation  write  out  24  terms  of  the  series  and  observe  that  the 
last  13  terms  destroy  each  other. 


ARITHMETICAL  PROGRESSION.  377 

Exercise  136. 

Find  the  sum  of : 

1.  5,  9,  13,  .....  to  19  terms. 

2.  10 J,  9,  7 J,  ....,  to  94  terms. 

3.  3  a,  a,  —  a,  . . . . ,  to  a  terms. 

4.  3 J,  2 J,  1|,  ....,  to  n  terms. 

^w— Im  —  2    m  —  3 

5.  ,  ,  ,  ....,  to  m  terms. 

m  mm 

.    2ag~l     ^  3    6  g^  -  5 

6.   ,  4a ,  ,  .....  to  n  terms. 

a  a  a 

^         4a  +  &    5a  +  2& 

7.  a,  — X — ,    ^ ,  ....,  to  19  terms. 

8.  The  first  term  is  3^,  and  the  sum  of  14  terms  is  84J. 
Find  the  last  term. 

9.  The  sum  of  40  terms  is  0,  and  the  common  difference 
is  —  ^.     Find  the  first  term. 

10.  Find  the  number  of  terms  and  common  difference: 

(1)  when  the  sum  is  24,  the  first  term  9,  and  tlie  last  —6; 

(2)  the  sum  49  a,  the  first  term  a,  and  the  last  13  a. 

11.  The  sum  of  12  terms  is   150,  and  the  first  is  5J. 
Determine  the  series. 

12.  Show  that  the  sum  of  the  first  n  odd  numbers  is  r?, 

13.  Find  the  sum  of  all  the  odd  numbers  between  100 
and  200. 


378  ELEMENTS  OF  ALGEBRA. 

14.  The  sum  of  five  terms  is  15,  and  the  difference  of 
the  squares  of  the  extremes  is  96.     Find  the  terms. 

15.  Find  the  sum  of  -=i,  tj ,  7=,  ...., 

l'\-  Vx    ^-^    1-  Vx 
to  n  terms. 

156.    a  is  called  the  arithmetical  mean  between  a  —  d  and  a  +  d. 
Hence, 

An  Arithmetical  Mean  is  the  middle  term  of  three  num- 
bers in  arithmetical  series. 

If  a  and  6  represent  two  numbers,  and  A  their  arithmetical  mean, 
the  common  difference  is  A  —  a,  or  b  —  A.     Therefore, 
A  —  a  —  h  —  A.     Hence  (solve  for  J.), 

To  Find  the  Arithmetical  Mean  Between  two  Terms. 

A  =  ^  M 

If  a  and  I  represent  any  two  numbers,  and  m  the  number  of  means 
between  them,  the  whole  number  of  terms  is  m  +  2,  or  wi  +  2  =  n. 
Substitute  this  value  for  n  in  (i)  (Art.  154), 

I  =  a  +  (m  +  I)  d.     Hence  (solve  for  d), 
To  Insert  any  Number  of  Arithmetical  Means  Between  two 

'^«"--  ^  .  1-  (V) 

This  finds  d,  and  the  m  required  means  are, 

a-\-  d,  a  +  2d,  a  +  ^d,  a  +  4Ld,  ....,  a-\-md. 

Example  1.     Find  the  arithmetical  mean  between:  (1)  27  and 

—  5 ;     (2)  rri^  -\-  mn  —  n^  and  m^  —  m  n  -\-  n^. 

Process.     (1)  Here,  a  =  27,    6  =  —  5. 

27-5 
Substitute  in  (iv),      A  =  — -x —  =  11. 

(2)  Here,  a  —  m^  +  mn-  n^,  b  =m^  —  mn  +  rfi. 

^  ,     .          .     ,.   ^          ,       m^  +  mn  --  n^  +  m^  —  mn  +  n^ 
Substitute  in  (iv),      A  = 2 — =;  m\ 


GEOMETlilCAL  TUOGRESSION.  379 

Example  2.     Insert  five  arithmetical  means  between  12  and  20. 
Process.     Here,   a  =  12,  /  =  20,  and  m  =  5. 

20-  12 
Substitute  in  (v),  d  =    ^      ^    =  H- 

The  series  is  12,  13^,  14|,  16,  17^,  18f,  20. 

Exercise  137. 
Insert : 

1.  14  arithmetical  means  between  —  7-J  and  —  2^. 

2.  16  arithmetical  means  between  7.2  and  —  6.4 

3.  10  arithmetical  means  between  5  m— 6  n  and  5w— 6  m. 

4.  4  arithmetical  means  between  —  1  and  —  7. 

5.  X  arithmetical  means  between  a^  and  1. 

«    ^.    ,    ,         .  ,        .    ,  ,  m— n       ,  m-hn 

6.  Find  the  arithmetical  mean  between  — — -  and . 

m+n         m — n 

7.  The  arithmetical  mean  between  two  numbers  is  —  9, 
and  the  mean  between  four  times  the  first  and  twelve  times 
the  second  is  —  66.     Find  the  numbers. 

GEOMETRICAL  PROGRESSION. 

157.  The  expressions  3,  9,  27,  81, ....,  and  1,  -i,  i,  -^,  ...., 
are  called  geometrical  progressions  or  series.  The  general  form  for 
such  a  series  is, 

a,  ar,  ar^,  ar*,  ar*,  ai*,  an*,  ar'',  ...., 

In  which  a  is  the  first  term,  and  r  a  constant  factor  or  ratio.    Hence, 

A  Geometrical  Progression  is  a  series  in  which  the  adja- 
cent terms  increase  or  decrease  by  a  constant  factor. 

The  Common  Ratio  is  the  fiictor  by  which  each  term  is 
multiplied  to  form  the  next  one. 


380  ELEMEJ^TS  OF  ALGEBRA. 

In  every  geometrical  series  the  following  elements  occur;  any 
three  of  which  being  given,  the  other  two  may  be  found. 

The  first  term,  or  a. 
The  last  term,  or  I. 
The  common  ratio,  or  r. 
The  number  of  terms,  or  n. 
The  sum  of  the  terms,  or  s. 

By  an  examination  of  the  general  form  it  is  seen  that  the  expo- 
nent of  r  is  always  1  less  than  the  number  of  the  term. 
Thus,  the  2d  term  is  a  r, 
3d  term  is  a  r^, 
4th  term  is  a  r*, 
12th  term  is  ar",  and  so  on. 
In  the  nth,  or  last  term,  the  exponent  of  r  is  w  —  1.     Hence, 

To  Find  the  Last  Term  of  a  Geometrical  Series,  when  the 

first  term^  the  comtnon  ratio,  and  the  number  of  terms  are 

given. 

I  =  ar""-^  (i) 

Notes:  1.   The  common  ratio  is  found  by  dividing  any  term  by  that  which 
immediately  precedes  it. 

2.  A  geometrical  series  is  said  to  be  increasing  or  decreasing,  according  as 
the  common  ratio  is  greater  than  1,  or  less  than  1. 

3.  An  arithmetical  series  is  formed  by  repeated  addition  or  subtraction;  a 
geometrical  series  by  repeated  multiplication. 

Example  1.     Find  the  8th  term  of  the  series  .008,  .04,  .2,  etc. 
Process.     Here,  a  =  .008,  n  =  8,  and  r  =  .04  ^  .008  =  5. 
Substitute  in  (i),  /  =  .008  X  58-i  =  625. 

Example  2.     Find  the  10th  term  of  — ,  x,  y,  — , 

Process.     Here,  a  =  — ,  n  =  10,  and  r  =  x  -. —  =  -  • 

y\  '  y     X 

Substitute  in  (i),  /  =  -  f|j  =  ar-'yS 


GEOMETRICAL  PROGRESSION.  381 

Exercise  138. 
Find: 

1.  The    5th  and    8th  terms  of  3,  6,  12,  .... 

2.  The  10th  and  16th  terms  of  256,  128,  64,  .... 

3.  The    8th  and  12th  terms  of  81,  -  27,  9,  .... 
4   The  14th  and    7th  terms  of  gJ^,  ^^,  3^,  .... 

.-    rr.1      «  1  ^  X    mx    m^x 

5.  The  6th  term  of  -,  — 5-,  — 3- 

y     /       f 

6.  The  mth  term  of  x,  x^^  7^,  .... 

7.  The  3d  and  6th  terms  are  f^  and  —  |.  Find  the 
series  and  the  12th  term. 

8.  The  5th  and  9th  terms  are  f  J  and  §.  Find  the 
series. 

9.  If  from  a  line  a  inches  in  length,  one  third  be  cut 
off,  then  one  third  of  the  remainder,  and  so  on;  what  part 
of  it  will  remain  when  this  has  been  done  5  times  ?  When 
/  times. 

168.    Taking  the  elements  as  given  in  Art.  157, 

8z=a  +  ar-\-ar^  +  ar*-\- -\-ai*-^-\-ar^'^  (1) 

Multiply  (1)  by  r, 

sr  =  ar-i-ar^-i-aH»-| -f- a  r"-'^ -I- g  r»-^ -t- g  r  (2) 

Subtract  (1)  from  (2), 

sr—8  =  ar^  —  a  (3) 

Substitute  the  value  of  or*  from  (i)  (Art.  167)  in  (3),  and  factor 
the  result,  «  (r  -  1)  =  r  /  -  a.     Hence  (solve  for  <), 

To  Find  the  Sum  of  all  the  Terms  of  a  Geometrical  Series 

8  =  ^^null  (ii) 

r  -  1 


382  ELEMENTS  OF  ALGEBKA. 

Example  1.     Find  the  6th  term  and  the  sum  of  —  J,  ^,  -  f ,  . 
Process.     Here,  a  =  —^,  7i  =  6,  and  r  =  —  |. 
Substitute  in  (i)  (Art.  157),  ^  =  -  |^  x  (-  |)^  =  |^. 

Substitute  in  (iii),  5  =  " '^_^s  _  ^  ^  =  W- 

EXxiMPLE  2.     Find  the  least  term  and  the  sum  of  3,  —  9,  27, 
to  7  terms. 

Process.     Here,  a  =  3,  n  —  1,  and  r  —  —  3. 
Substitute  in  (i)  (x\rt.  157),  Z  =  3  (-  3)«  =  2187. 

3^:^y— =1641. 


Substitute  in  (ii), 

^=      -3 

Exercise  139. 

Find  the  sum  of: 

1.    3,  -1,  J,.... 

,  to  6  terms. 

2.  -lh-i>' 

,...,  to  6  terms. 

3.  1,  -J,  A.  •• 

. .,  to  8  terms. 

1            3 
*•  V3'    '  V3' 

....,  to  8  terms. 

5.    1,  3,  32,  ..... 

to  m  terms. 

6.   2,  -4,  8,  ... 

. ,  to  2  m  terms. 

7.  The  7th  and  4th  terms  are  625  and  —  5.  Find  the 
1st  term,  and  the  sum  of  the  4th  to  the  7th  terms  inclusive. 

8.  The  sum  of  the  first  10  terms  is  equal  to  33  times 
the  sum  of  the  first  5  terms.     Find  the  common  ratio. 

9.  The  sum  of  three  numbers  in  geometrical  progression 
is  216,  and  the  first  term  is  5.  Find  the  common  ratio 
and  the  numbers. 


GEOMETRICAL  PliOGRESSION.  383 

159.     A  Geometrical  Mean  is  the  middle  term  of  three 
numbers  in  geometrical  series. 

If  a  and  b  represent  two  numbers,  and  G  their  geometrical  mean, 

G         b 
the  common  ratio  is  — ,  or  ^.     Therefore, 

G      b 

—  =  jy.    Hence  (solve  for  G), 

To  Find  the  Geometrical  Mean  Between  two  Terms 

O  =  \/ab  '  (iv) 

If  a  and  b  represent  any  two  numbers,  and  m  the  number  of  means 
between  them,  the  whole  number  of  terms  is  m  +  2,  or  m  -f-  2  =  n. 
Substitute  this  value  for  n  in  (i)  (Art.  167), 

/  =  a  r"*  + 1.     Hence  (solve  for  r), 

To  Insert  any  Number  of  Geometrical  Means  Between  two 
Terms.  ,  ,  _ 

This  finds  r,  and  the  m  required  means  are, 

ar,  ar^,  ar^,  ar^,  ar^ ,  ar»*. 

Example  1.     Find  the  geometrical  mean  between  :   (1)  — -=  and 
3  V3 

7^;   (2)  3x»y  and  I2xfz. 

"^^  1  3 

Process.     (1)  Here,  a  =  -—p,  and  b  =  —-z  • 


Substitute  in  (iv),       G  =  v/ — p  ^  ^=^ 


3^ 

V3       V3 

(2)  Here,  a  =  3x*y  and  b  =  Uxy'z. 

Substitute  in  (iv),       G  =  ^33*y  X  I2xy*z  =  6x^y^ ^z. 

Example  2.     Insert  six  geometrical  means  between  14  and  -  ^y. 

Process.     Here,  a  =  14,  /  =  -  /y,  and  m  =  6. 

Substitute  in  (v),  r  =  ^—^\^  =  —  i- 

Hence,  the  series  is  14,  -7,  f  -  },  |,  -^^h-  ^. 


384  ELEMENTS  OF  ALGEBRA. 

Exercise  140. 

Find  the  geometrical  mean  between : 

1.  7  and  252;    a^h  and  ah^-,   f  and  |i ;  |  and  ||. 

2.  yV  '^"cl  jJ^o  ;  4^:2  -  12a;  +  9  and  ^x^-\-12x^-  4. 
Insert : 

3.  2  geometrical  means  between  5  and  320. 

4.  2  geometrical  means  between  1  and  \. 

5.  3  geometrical  means  between  100  and  2J|. 

6.  6  geometrical  means  between  14  and  —  -^^. 

7.  7  geometrical  means  between  2  and  13,122. 

8.  Which  is  the  greater,  and  how  much  greater,   the 
arithmetical  or  geometrical  mean  between  1  and  \. 

9.  Find  two  numbers  whose  sum  is  10,  and  whose  geo- 
metrical mean  is  4. 

HARMONICAL  PROGRESSION. 

160.    The  expressions  h  ^,  \,  \,  ....,  and  4,  -f,  -  f,  -4,  ...., 
are  called  harmonical  progressions  or  series,  because  their  reciprocals 

1,  3,  5,  7, ,  and  ^,  —  |,  —  |,  —  |., .  form  arithmetical  series. 

The  general  form  for  such  a  series  is, 

I-   ^>   Wh^ ^TT^^ry-.-     Hence, 

An  Harmonical  Progression  is  a  series  the  reciprocals  of 
whose  terms  form  an  arithmetical  series. 


HARMONICAL  PROGRESSION.  385 

Notes:  1.  Evidently  all  questions  relating  to  harmonical  pro^^ression  are 
readily  solved  by  writing  the  reciprocals  of  the  temis  so  as  to  form  an  arith- 
metical series. 

2.  There  is  no  general  formula  for  finding  the  sum  of  the  terras  of  a  har- 
monical series. 

3.  The  term  harmonical  is  derived  from  the  fact  that  musical  strings  of 
•qual  thickness  and  teiisiou  produce  harvumy  when  sounded  together,  if  their 
lengths  are  represented  by  the  reciprocals  of  the  series  of  natural  numbers;  that 
is,  by  the  series  1,  J,  J,  i,  J,  J,  etc.  Harmonical  properties  are  also  interesting 
because  of  their  importance  in  geometry. 

Example  1.     Find  the  mth  term  of  the  series  3,  IJ,  1,  f ,  f ,  etc. 
Solution.    Taking  the  reciprocals  of  the  terms,  we  have  ^,  |,  1, 
|,  |,  etc. ;  an  arithmetical  series. 
Here,  a  =  |,  rf  =  J,  and  n  =  m. 

Substituting  in  (i)  (Art.  154),  d  =  ^  +  {m  -  I)  ^  =  ^.     Taking 

the  reciprocal  of  this  value  for  the  required  term,  we  have  — . 

Example  2.  The  12th  term  is  |,  and  the  19th  term  is  ^.  Find 
the  series. 

Process.  The  12th  and  19th  terms  of  the  corresponding  arith- 
metical series  are  5  and  ^. 

From  (i)  (Art.  154),  5  =  o  +  11  rf, 
^  =  a  +  18  d. 

Solving  for  a  and  rf,   a  =  |  and  rf  =  J. 

The  arithmetical  series  is,  |,  |,  2,  t,  |,  3,  Y>  •  •  •  • 

The  harmonical  series  is,    |,  |,  ^,  ^,  |,  J,  i%, 

161.  A  Harmonical  Mean  is  the  middle  term  of  three 
numbers  in  harmonical  series. 

If  a  and  b  represent  two  numbers,  and  H  their  harmonical  mean, 
the  corresponding  arithmetical  series  is  -,  ^,  ^.     The  common  dif- 

.11        !!,«,. 
ference  is  ^  ~  o*  ^^  6  ~  H'    ^*^^'^'®^» 

D-  —  -  =  T  -  ^.     Hen'ce  (solve  for  //), 

25 


386  ELEMENTS   OF  ALGEBRA. 

To  Find  the  Harmonical  Mean  Between  two  Numbers. 

H  -  ^^^  (i) 

Example  L     Find  the  harmonical  mean  between :  (1)  ^  and  ^  ; 
(2)  X  -j-  y  and  x  —  y. 

Process.     (1)  Here,  a  =  \  and  b  —  ^j^. 

Substitute  in  (i),  H  =  \ 

(2)  Here,  a  =  x  -\-  y  and  b  =  x  —  y. 

X^  -  7/2 

Substitute  in  (i),  H— '—  . 


Example  2.     Insert  three  harmonical  means  between  f  and  ^^. 
Process.     The  terms  of  the  corresponding  arithmetical  series  are 
I  and  J^. 

Here,  a  =  |,  I  =  ^^,  and  m  =  3. 
Substitute  in  (v)  (Art.  156),  d  =  ^. 
The  three  arithmetical  means  are      ^,  ^,  ^. 
The  required  harmonical  means  are  y\,  f ,   ^. 


Exercise  141. 

1.  Find  the  8th  term  of  IJ,  l^f  2-^2^,  .... 

2.  Find  the  21st  term  of  21    llf,  1^^,  .... 

3.  The  39th  term  is  y^-,  and  the  54th  term  is  ^.     Find 
the  series. 

4.  The  2d  term  is  2,  and  the  31st  term  is  ^*y.     Find  the 
first  six  terms. 

Insert : 

5.  One  harmonical  mean  between  1  and  13. 

6.  3  harmonical  means  between  2f  and  12. 


HARMONICAL  PROGRESSION.  387 

7.  4  harmoiiical  iiu'iiii.s  buLu  uuii  |  iind  ^. 

8.  6  harinouical  means  between  3  and  ^. 

9.  The  arithmetical  mean  of  two  numbers  is  9,  and  the 
harmonical  mean  is  8.     Find  the  numbers. 

10.  The  difference  of  the  arithmetical  and  harmonical 
means  between  two  numbers  is  1.  Find  the  numbers;  one 
being  three  times  the  other. 

11.  Find  two  numbers  such  that  the  sum  of  their  arith- 
metical, geometrical,  and  harmonical  means  is  9|,  and  the 
product  of  these  means  is  27. 

12.  The  arithmetical  mean  between  two  numbers  ex- 
ceeds the  geometrical  by  2^,  and  the  geometrical  exceeds 
the  harmonical  by  2.     Find  the  numbers. 

13.  The  sum  of  three  terms  of  a  harmonical  series  is  37, 
and  the  sum  of  their  squares  is  469.     Find  the  numbers. 

14.  The  sum  of  three  consecutive  terms  in  harmonical 
series  is  1^,  and  the  first  term  is  J.     Find  the  numbers. 

15.  Arrange  the  aritlimetical,  geometrical,  and  harmoni- 
cal means  between  two  numbers  a  and  h  in  order  of 
magnitude. 

16.  If  50  potatoes  are  placed  in  a  line  3  feet  from  each 
other,  and  the  first  is  3  feet  from  a  basket,  how  far  will  a 
person  travel,  starting  from  the  basket,  to  gather  them  up 
singly,  and  return  with  each  to  the  basket  ? 

17.  There  are  four  numbers  in  geometrical  progression, 
the  first  of  which  is  less  than  the  fourth  by  21,  and  the 
difference  of  the  extremes  divided  by  the  difference  of  the 
means  is  equal  to  3 J.     Find  the  numbers. 


388  ELEMENTS  OF  ALGEBRA. 


CHAPTEK  XXVII. 
RATIO  AND   PROPORTION. 

162.  The  Ratio  of  two  numbers  is  their  relative  magni- 
tude, and  is  expressed  by  the  fraction  of  which  the  first  is 
the  numerator  and  the  second  the  denominator. 

Thus,  the  ratio  of  10  to  5  is  expressed  by  the  fraction  ^-^  ;  the 
ratio  of  I  to  I  is  expressed  by  the  fraction  f  -r  f  (=  y%). 

The  ratio  of  two  quantities  of  the  same  kind  is  equal  to  the  ratio 
of  the  two  numbers  by  which  they  are  expressed. 

Thus,  the  ratio  of  $5  to  $6  is  | ;  of  15  apples  to  3  apples  is  i/ ;  of 
3f  feet  to  5^  feet  is  3f  ~  5|  =  ^|. 

,    The  Sign  of  ratio  is  the  colon  :,  -^,  or  the  fractional 
form  of  indicating  division. 

a 
Thus,  the  ratio  of  a  to  6  is  expressed  by  a  :  b,  or  a  -f  6,  or  t,  any 

one  of  which  may  be  read  "a  is  to  6/'  or  "ratio  of  a  to  &." 

The  Terms  of  a  ratio  are  the  numbers  compared.  The 
first  term  is  called  the  antecedent,  the  second  the  conse- 
quent, and  the  two  terms  together  are  called  a  couplet. 

A  ratio  is  called  a  ratio  of  greater  inequality,  of  less 
inequality,  or  of  equality,  according  as  tlie  antecedent  is 
greater  than,  less  than,  or  equal  to,  the  consequent. 

An  Inverse  Ratio  is  one  in  which  the  terms  are  inter- 
changed ;  as,  the  ratio  of  7  :  8  is  the  inverse  of  the  ratio 
8:7. 

A  Compound  Ratio  is  the  product  of  two  or  more  simple 
ratios;  as,  the  compound  ratio  2  :  3,  5  :  4,  15  :  6,  is  150  :  72. 


RATIO  AND  PROPORTION.  389 

NotM  :  1.  A  quantity  may  be  detiaed  as  a  definite  portion  of  any  magni- 
tude. Thus,  any  definite  number  of  dollars,  poimds,  bushels,  acres,  feet,  yards, 
or  miles,  is  a  quantity. 

2.  To  compai*e  two  quantities  they  must  be  expressed  iu  terms  of  the  same 
unit.    Thus,  the  ratio  of  2  rods  to  9  inches  is  expressed  by  the  fraction, 
16J  X  2  X  12      396 


163.  Evidently  the  ratios  4  :  5,  8  :  10,  ^  :  Yi  are  equal  to  each 
other.     Ill  general, 

a      ma       „ 

I.  If  the  terms  of  a  ratio  are  multiplied  or  divided  by 
the  sanu  number,  the  value  of  the  ratio  is  not  changed. 

The  ratio  9  :  7  is  compared  with  the  ratio  4  :  3  by  comparing  ^ 
and  |.  ^  =  ^\,  and  |  =  ^|.  Therefore,  4  :  3  is  greater  than  9  •  7. 
Hence, 

II.  Ratios  are  compared  by  comparing  the  fractions  that 
represent  them. 

If  to  each  term  of  the  ratio  5  :  4  we  add  16,  the  new  ratio,  21  :  20, 
is  less  than  the  ratio  5  :  4,  because  \  is  greater  than  f  ^.  If  to  each 
tenn  of  the  ratio  4  :  5  we  add  16,  the  new  ratio,  20  :  21,  is  greater 
than  the  ratio  4  :  5.     Hence, 

III.  A  ratio  of  greater  iiuquality  is  diminished,  and  a 
ratio  of  less  inequality  is  increased,  by  adding  the  same 
number  to  both  its  terms. 

If  from  each  term  of  the  ratio  32  :  30  we  subtract  24,  the  new 
ratio,  8  :  6,  is  greater  than  the  ratio  32  :  30.  If  from  each  term  of 
the  ratio  28 :  30  we  subtract  16,  the  new  ratio,  13  :  15,  is  less  than 
the  ratio  28  :  30.     Hence, 

IV.  A  ratio  of  greater  inequality  is  increased,  and  a 
ratio  of  less  inequality  is  diminished,  by  taking  the  same 
number  from  both  terms. 


390  ELEMENTS  OF  ALGEBRA. 


a       c       e      g 
Suppose   ^  =  ^=^=^  =  r. 

Simplify,  br  —  a,  dr  =  c,  fr  —  e^  hr  =  g. 
Add  the  corresponding  members  and  factor  the  result, 
{b-\-d+f+h)r  =  a  +  c  +  e-\-g. 

Therefore,  ^  =  6  +  ^  t/t  I  =  ^  =  "^  =}"  f'    H"^'"» 

V.  In  a  series  of  equal  ratios,  the  sum  of  the  antece- 
dents divided  hy  the  sum  of  the  consequents  is  equal  to  any 
antecedent  divided  by  its  consequent. 

Notes :  1.  The  sign  : ,  is  an  exact  equivalent  for  the  sign  of  division ;  and  is 
a  modification  of  -r  . 

2.  A  Duplicate  Batio  is  the  ratio  of  the  squares ;  a  Triplicate,  of  the  cubes ; 
a  Subduplicate,  of  the  square  roots  ;  a  Subtriplicate,  of  the  cube  roots  of  two 
numbers.  Thus,  a^  :  b^ ;  a^  :  b^  ;  Va  :  Vb;  fa  :  Vb  are  respectively  fhe 
duplicate,  triplicate,  subduplicate,  and  subtriplicate  ratios  of  a  to  b. 

Example  1.     Find  the  ratio  compounded  of  the  duplicate  ratio  of 
2  a    a2      _ 
-T-  :  T2  V 6,  and  the  ratio  3ax  :  2by. 

Process,      ihe  duplicate  ratio  oi  -7-  :  Tg  yB  is  -r^-  :  -r^  • 

,       .     4a2    6a<                              I2a^x    ]2a*by 
Ihe  compound  ratio  -vg-  :  -p- ?  ^ax  :  2by,is  — p—  : 74 —  • 

I2a^x     \2a^by       ISa^x       12  a^hy      bx 

But  — To —  : TT— ^  =  — To i u —  =  —  =  bx  :  ay. 

b^  6*  b^  b^  ay  ^ 

Example  2      If  15  (2  2-2  —  y'^)  —  ^  xy,  find  the  ratio  x  :  y. 
Process.     From  the  given  equation,  a:^  —  jV  a:  ?/  =  ^  2/^. 
Complete  the  square  and  solve  for  x,  a;  =  ^  ?/,  or  —  f  i/* 

X 

Therefore,  -  =  a  or  - 1 . 

Exercise  142. 

Find  the  ratio  compounded  of : 

1.  The  ratio  2  a  :  3 &,  and  the  duplicate  ratio  oi^b'^'.a'h. 

2.  The  subduplicate  ratio  of  64  :  9,  and  the  ratio  27  :  56. 


RATIO.  391 

3.  The  duplicate  ratio  of  4  :  15,  and  the  triplicate  ratio 
of  5  :  2. 

4.  1  —  3^  :  1  +  y,  X  —  X  i/  :  I  +  s^,  and  1  :  x  —  x^. 
a-\-h  g'+fe^  {a^-l^f  a2_9^^2Q   ,       a8-13a  +  42 

Simplify  each  of  the  ratios  : 

6.  5ax:4:x;    li5xy:20a^;    2x^y:\x^. 

7.  iaxy.z^ay^;  -^ ^ :a^na^. 

Arrange  the  following  ratios  in  order  of  magnitude  : 

8.  5  :  6,  7  :  8,  41  :  48,  and  31  :  36. 

9.  a  ^  b  :  a  -\-  bf  and  a^  —  I?  :  a^  +  b^,  when  a  >  6. 

10.  For  what  value  of  x  will  the  ratio  15  +  a::17  +  ic 
be  equal  to  the  ratio  1:2? 

11.  Find  X  \  yy  if  a:^  +  6  ?/2  =  5  a;  y. 

12.  Find  the  ratio  of  x  to  y,  if  the  ratio  4a;  +  5y  :  3a;— y 
is  equal  to  2. 

13.  What  number  must  be  added  to  each  term  of  the 
ratio  a  :  h,  that  it  may  become  equal  to  the  ratio  m  :  nl 

14.  What  number  must  be  subtracted  from  the  conse- 
quent of  the  ratio  a  :  b,  that  it  may  become  equal  to  the 
ratio  m  :  7l1 

15.  A  certain  ratio  will  be  equal  to  2  :  3,  if  2  be  added 
to  each  of  its  terms;  and  it  will  be  equal  to  1  :  2,  if  1  be 
subtracted  from  each  of  its  terms.     Find  the  ratio. 


392  ELEMENTS  OP  ALGEBRA. 

16.  If  a  :  6  be  in  the  duplicate  ratio  oi  a  -\-  x  :  h  +  x, 
find  X. 

17.  Show  that  a  duplicate  ratio  is  greater  or  less  than 
its  simple  ratio,  according  as  it  is  a  ratio  of  greater  or  less 
inequality. 

PROPORTION. 

164.  A  Proportion  is  an  equality  of  ratios.  Four  num- 
bers are  in  proportion,  when  the  first  divided  by  the  second 
is  equal  to  the  third  divided  by  the  fourth. 

a       c 
Thus,  if  T  =  -7 ,  then  a,  h,  c,  d,  are  called  proportionals,  or  are  said 

to  be  in  proportion,  and  they  may  be  written  in  either  of  the  forms : 

a  :  b  ::  c  :  d, 
read,  "a  is  to  &  as  c  is  to  d! ;" 
or  a  :  b  =  c  :  d, 

read,  "the  rat'o  of  a  to  &  is  equal  to  the  ratio  of  c  to  d;" 

a       c 
^^  b=d^ 

read,  "a  divided  by  b  equals  c  divided  by  d." 

The  Terms  of  a  proportion  are  the  four  numbers  com- 
pared. The  first  and  third  terms  are  called  the  antecedents, 
the  second  and  fourth  terms,  the  consequents;  the  first  and 
fourth  terms  are  called  the  extremes,  the  second  and  third 
terms,  the  means. 

Thus,  in  the  above  proportion,  a  and  c  are  the  antecedents,  b  and 
d  the  consequents,  a  and  d  the  extremes,  b  and  c  the  means. 

Note  1.  The  algebraic  test  of  a  proportion  is  that  the  two  fractions  which 
represent  the  ratios  shall  he  equal. 


PROPORTION. 

Let 

a:b  ::  c  :d. 

By  definition, 

a       c 
~b^d' 

Free  from  fractions, 

ad  =  bc.    Hence, 

393 


I.  In  any  proportion  the  product  of  the  extremes  is  equal 
to  the  product  of  the  means. 

Note  2.  If  any  three  terms  in  a  proportion  are  given,  the  fourth  may  be 
found  from  the  relation  that  the  product  of  the  extremes  is  equal  to  the 
product  of  the  means. 

Let  ad  =  be.  \ 

a       c 
Divide  by  6 rf,  b~d' 

By  definition,  a:b::c:d.     Hence, 

IL  If  tJu  product  of  tivo  numbers  is  equal  to  the  pro- 
duct of  two  others,  either  two  may  he  made  tlie  extremes 
of  a  proportion  and  the  other  two  the  7neans. 

A  Mean  Proportional  is  a  number  used  for  both  means 
of  a  proportion  ;  as,  h,  in  the  proportion  a  :b  ::h  :  c. 

A  Third  Proportional  is  the  fourth  term  of  a  proportion 
in  which  the  means  are  equal;  as,  c,  in  the  proportion 
a  :  h ::  h  :  c 

Ia'I  a  :  b  ::  b  :  c. 

Therefore!.,  b^  =  ac. 

Extract  the  square  root,        b  =  \/a  c.     Hence, 

IIL  A  mean  proportional  between  two  numbers  is  eqv^ 
to  the  square  root  of  their  product. 

Let  a  :b  ::  c  :  d. 

Therefore  1.,  ail  —  he, 

a       b 
Divide  by  c rf,  ~  ~  d' 

By  definition,  a.  cy.b  .  d.     Hence, 


394  ELEMENTS  OF  ALGEBRA. 

IV.  If  four  numhers  are  in  proportion,  they  will  he  in 
proportion  hy  alternation;  that  is,  the  first  will  he  to  (he 
third,  as  the  second  is  to  the  fourth. 

Let  a  :b  ::  c  :  d. 

Then  I.,  bc  =  ad. 

Divide  by  a  c,  -::=_. 

•^  a      c 

By  definition,  b  :  a  ::  d  :  c.     Hence, 

V.  If  four  numhers  are  in  proportion,  they  will  he  in 
proportion  hy  inversion;  that  is,  the  second  will  he  to  the 
first  as  the  fourth  is  to  the  third. 


Let                                      a:b::c:d. 

By  definition,                         h~  d' 

a            c 
Add  1  to  each  member,  t+  1  =  ^  +  1, 

a  +  b      c-^d 
b     ~     d 
Therefore,                    a  +  b  :  b  ::  c  +  d  :  d. 

Hence, 

VI.  If  four  numhers  are  in  proportion,  they  will  he  in 
proportion  hy  composition;  that  is,  the  sum  of  the  first 
two  will  he  to  the  second  as  the  sum  of  the  last  two  is  to 
the  fourth. 

Let  a  :  b  ::  c  :  d. 

.  a       c 

By  definition,  J  ~  d' 

Subtract  1  from  each  member, 

a  c 

a~b      c— d 

«^  -~r  =  ~d"- 

Therefore,  a  —  b:b::c-d:d.     Hence 


PROPORTION.  395 

VII.  Jf  four  niimhers  are  in  propoi'tion,  they  will  he 
in  proportion  hy  division ;  that  is,  the  difference  of  the  first 
two  will  be  to  the  second  as  the  difference  of  the  last  two 
is  to  the  fourth. 

Let  a  :l  ::c  '.d. 

a  4-h      c  +  d 
Then  VI.,  =-|  ^     • 

also  VII., 


c 
a  —  b      c  —  d 


Divide, 


b  c 

a+b      c+d 


a  —  b      c  —  d 
By  definition,      a  +  b  :  a  —  b  ::  c  +  d  :  c  —  d.    Hence, 

VIII.  If  four  numbers  are  in  proportion,  they  will  he 
in  proportion  hy  composition  ajid  division ;  tliat  is,  the  sum 
of  the  first  two  will  he  to  their  differeiue  as  the  sum  of 
the  last  two  is  to  their  difference. 

Let  a  :b'.:c  :d, 

e:f::g:h, 

k  :  I  ::  m  :  n. 
T»,/...        ^      c    e      g    k      m 
By  definition,   l  =  d'f=h'l=n' 

Multiply  the  corresponding  members  of  the  equations  together, 

aek      c  gm 
Ff'l'^dir^' 
By  definition,  a  e k  :  b  f  I  ::  c g m  :  d  h  n.     Hence, 

IX.  The  products  of  the  corresponding  terms  of  two  or 
more  proportions  are  in  proportion. 

Let  a:b  ::  c  :  d. 

d      c 
By  definition,  h~  d' 

Raise  each  member  to  the  nth  power,  fcS  ~  ^  * 


396  ELEMENTS  OP  ALGEBRA. 

Therelore,  a"  :  6"  : :  c"  :  d\ 

1         1 

Extract  the  nth  root  of  each  member,  —  =  -^  • 

Therefore,  a"  :  6^  ::  c"  :  cT.     Hence, 

X.  /?i  a^iy  proportion  like  'powers  or  like  roots  of  the 
terms  are  in  proportion, 

A  Continued  Proportion  is  a  series  of  equal  ratios ; 

As,  8:4::12:6::10:5  ::16:8;  a  :  b  ::  c  :  d  ::  e  :f::g:h,  read 
"a  is  to  6  as  c  is  to  d  a.s  e  is  to/  as  g  is  to  A." 

Kote  3.  Four  numbers  are  said  to  foiin  a  continued  proportion  when  each 
consequent  is  the  antecedent  of  the  next  ratio ;  as,  a  :  b  ::  b  :  c  ::  c  :  d. 

Let  a  :b  ::  c  :  d  ::  e  :/  ::  g  :h. 

a       c      e      g 
By  definition,  ^  =  ^^  =  ^=:  ^^  • 

a  -^c  -^  e  +  q      a       c      e      g 
ByV.,  (Art.  163),  l^d-v  f+h^  h^  d  =  r'h 

Therefore,  a-\-c-\-€  +  g'.h  +  d  +f  -hh  i:  a  :  b.     Hence, 

XI.  In  a  continued  proportion  the  sum  of  the  ante- 
cedents is  to  the  sum  of  the  consequents  as  any  antecedent 
is  to  its  consequent. 

^^     a2+62       ab  +  bc 

Example  1.     If   -  .— r  i>  ^  =  ~"a2~i — 2  »   prove  that  6  is  a  mean 
ab  +  be        6^  4-  c^  '    '■ 

proportional  between  a  and  c. 

Proof.  Free  the  given  equation  from  fractions,  transpose  and 
factor,  (b^  —  acy  —  0,  or  b^  =  ac. 

Therefore  II.,  a  :  h  ::  b  :  c. 

Example  2.  If  a  .  b  ::  c  :  d,  prove  that  m  a'^  +^p  b'^+ nab  :  mc^ 
-i-pd^  +  ncd  ::  b^  :  d^. 


iVI., 

a      b 

c~  d 
a«      ab 
7^=7d' 
a*      b» 

ab 

nab 

a^      n  (t  li 

cd- 

ncd' 

or 

c2  ~  'nc  il 

b^ 

pb^ 

a^      pb^ 

d^' 

~  pd^' 

or 

c^~  pd^ 

PROPORTION.  397 

a      b         ^1 V 
Proof.     From  the  giveu  proportion  VI.,  c~  7l  ^ 

a 
Multiply  by  - , 

Square  Loth  members  of  (1), 

By  I.  (Art.  163), 

Also  I.  (Art.  163), 

Also  I.  (Art.  163),  r»  =  ^a- 

m a*      pb^  _  n  ab  _  b'^      a' 
Hence,  ^^  =^  =  ^^-^  =  ^  = -^. 

By  V.  (Art.  163),  ^...^'pd^^ncd  =  d^^' 

Therefore  XL,  ma^  +  pb^ -^-na  b  :mc^  +  pd^+  ncd  ::  b^  :  d^. 

Example  3.     Find  x  when  ^m-\-x-\-  ^m-x  -.  ^m-^x-^m—x 
::n  ;  1. 

Procefw.     By  VIII.,     2  ^m-\-x  :  2  ^m-x  ::  n+l  :  n-1, 

or  I.  (Art.  163),  /^/^Ta:  :  ^m=^  ::  n+l  :  n-1, 

By  X.,  m  +  x  :  m  -x  ::  (n+l)»  :  (n-l)«. 

By  I.,  {m  +  x)in-  1)»  =  (m-x)(n-M)«. 

Simplify,  transpose,  and  factor,   2  n  {n^  +  3)  x  =  2  m (3  nH  1). 

_m(3n«  +  l) 
Therefore,  x  -   ^(^8^.3)  • 

Exercise   143. 

J(  ad  =  bc,  prove  that : 

1.  d  :b  ::  c  :  a;    d  :  c  ::b  :  a;    h  :  a  ::  d  :  c. 

2.  hidr.a'.e;    c:a::d:h;    c:d::a:h. 


398  ELEMENTS   OF  ALGEBRA. 

Find  a  mean  proportional  between : 

3.  2  and  8;  3  and  1 J  ;  Handf;  8  and  18;  a^bsmdah^. 

4.  (a  +  hf  and  (a  -  hf ;    360  a*  and  250  d^  h\ 
Find  a  third  proportional  to  : 

5.  I  and  f ;  I  and  | ;  .2  and  .4 ;  2  and  3  ;    f  and  f 

6.  1  and  VI;   (a  -  hf  and  a^  -  62;  ?  +  ?^  and  -  • 
Find  a  fourth  proportional  to  : 

7.  2,  5,  and  6 ;  4,  |,  and  | ;  f ,  f ,  and  f  ;    a,  ak,  and  6. 

8.  a3,  ah,  and  5a22,.    ___,  _______  and  ^g--^ 

If  a  :  &  ::  c  :  6?,  prove  that: 
9.    a  +  h  :  a  :\  c  -\-  d  :  c\    a  —  h  \  a  w  c  —  d  :  c. 

10.  ac  :b  d  ::  c^  :  d!^;    ab  :  cd  ::  a^  :  c^. 

11.  2a  +  3c:3a  +  2c::26+3f^:3&+2c?. 

12.  3  a  -  5  6  :  3  c  —  5  c?  ::  5  a  +  3  6  :  5  c  +  3  c^. 

^oo         A7       o        ^7^^  +  ^       r^,  —  6       a       b 

13.  fa:|?.::|.:H;    -^  =  __^  .  _  =  ^  . 

U.    3  a  +  2b  :  S  a  -  2b  ::  3  c  -{-  2d  :  3  c  -  2  d. 

15.  la  +  77ib  :  pa  +  qb  ::  I  c  +  md  :  p  c  +  qd.' 

16.  ft3  .  2,3  ..  ^3  .  ^3.     ^2  .  ^2  ..  ^2  _  ^2  .  ^2  _  ^2 

17.  a2  +  c2  :  a&  +  C6^  ::  a6  +  C6?  :  b^  +  ^2. 

18.  V^r:r^:V6::VM^:  V^;    l  =  \^P^' 


PROPORTION.  399 

If  6  is  a  mean  proportioual  between  a  and  c,  prove  that; 

If  «  :  6  ::  c  :  rf  ::  6  :/,  prove  that : 

on         J.        .     i      7.i^_L/    O'+'^c-^Se       2a4-3c+4g 

20.  a:fe::a  +  .  +  .:^  +  ^+/;^^2rf^3/=26+3rf+4/' 

21.  rf  is  a  third  proportional  to  a  and  6,  and  c  is  a  third 
proportional  to  b  and  a,  find  a  and  6  in  terras  of  d  and  c. 

li  m  +  n  :  m  —  n  ::  X  -\-  y  :  X  —  y,  prove  that  : 

22.  ^  •\'  w?  \  ^  —  iv?  \\  y^  ■\-  n^  \  y^  ^  n^. 
Solve  the  following  proportions : 


23. 

24. 

a:3  ~  yS  :  (a;  -  y)8  ::  19  :  1  and  a;  :  6  ::  4  :  y. 

25. 

If = =     ,  show  that  a  +  b-{-c  =  0. 

X  —  y      y  —  z      z  —  X 

26.  A  and  B  engage  in  biisine^Jj  with  different  sums. 
A  gains  SloOO,  B  loses  $500,  after  which  A's  money  is  to 
B's  as  3  to  2 ;  but  had  A  lost  $500  and  B  gained  $1000, 
then  A's  money  would  have  been  to  B's  as  5  to  9.  Find 
each  man's  investment. 

27.  Show  that  the  geometrical  mean  is  a  mean  propor- 
tional between  the  arithmetical  and  harmonical  means 
between  the  two  numbers  a  and  b, 

28.  When  «,  &,  c,  are  in  harmonical  progression,  show 
that  a:c::a  —  5:&  —  c.  Hence,  of  three  consecutive 
terms  of  a  harmonical  series,  the  first  is  to  the  third  as  tht 
first  minus  the  second  is  to  the  second  minm  the  thirds 


400  ELEMENTS  OF  ALGEBRA. 

29.  Find  the  ratio  compounded  of  the  ratio   3  «  :  4  6, 
and  the  subduplicate  ratio  of  16  &*  :  9  a* 

30.  If  -  =  3i,  find  the  value  of  ^  ~  '^  ^  - 


y  '  2x-  by 

31.  If  6  :  a  : :  2  :  5,  find  the  value  of  2a-Zh:Zh—a. 

32.  If  T  =  T,  and  -  =  ^,  find  the  value  of  ^7 ^^  • 

6      4  2/       '  4:hy  —  lax 

33.  If    7  m  —  4  71  :  3  wt  +  ii  : :  5  :  13,   find   the   ratio 
m  :  n. 

34    If s s—  =  -TT ,  find  the  ratio  w  :  n. 

m2  +  7l2  41 

35.    If  2  a:  :  3  y  be  in  the  duplicate  ratio  of  2  a?  —  m  :  3  ?/ 
—  m,  find  the  value  of  m. 


a       c       m 


36.    If  -  =  -  =  -,  prove  that  each  of  these  ratios  is 
equal  to  ^WHH^L 


4  m^c 


37.  If  2a  +  3Z»  :  2,a  -  36  ::  2^2+  37i2  :  2^2-  3ii2, 
show  that  a  has  to  h  the  duplicate  ratio  that  m  has  to  7i. 

38.  A  railway  passenger  observes  that  a  train  passes 
him,  moving  in  the  opposite  direction,  in  30  seconds ;  but 
moving  in  the  same  direction  with  him,  it  passes  him  in 
90  seconds.     Compare  the  rates  of  the  two  trains. 

Solve  the  following  proportions  : 

39.  Vx  +  Vh  :  Vx  -  Vh  '.:  a  :h)  2"^'  :  22-  ; :  8  :  1. 


(x  +  y-.x 


:  X  —  y  '.'.  m  +  n  \  m  —  n* 
40.  ^    „         o        2         2   .      2      1 


APPENDIX, 


COMPUTATION  OF  LOGARITHMS. 

Since  the  logarithms  of  all  composite  numbers  are  found  by  add- 
ing the  logarithms  of  their  factors  (Art.  122),  it  is  only  necessary  to 
compute  the  logarithms  of  prime  numbei-s. 

The  following  method  for  computing  logarithms  is  the  one  that 
was  used  when  our  tables  were  first  made,  although  it  is  not  the  most 
expeditious  method  now  known. 

Example  1.    Find  the  logarithm  of  5. 
Since  10«>  =    1, 

and  101  =  10  (1) 

and  as  5  lies  between  1  and  10,  its  logarithm  must  lie  between  0  and  1. 

Extract  the  square  root  of  (1),        10-6  =  3. 162277+  (2) 

As  5  lies  between  10  and  3.1622774-  its  logarithm  lies  between 
1  and  .5. 

Multiply  (2)  and  (1)  together,       I0i»  =  31.62277  f. 

Take  the  square  root,  10 '»  =    5.6234134  (3) 

5  lies  between  3.162277+  and  5  623413+ ,  and  its  logarithm  must 
lie  between  .5  and  .75. 

Multiply  (2)  and  (3)  together,      lO^-^  =  17.7827895914+. 

Take  the  square  root,  10«26  =    4.216964+  (4) 

Since  5  lies  between  5.623413+  and  4.2 16964+ ,  its  logarithm 
must  lie  between  .75  and  .625. 

Multiply  (3)  and  (4)  together,  take  the  square  root  of  the  result, 
and  we  have  1()«876  _  4.869674+.  Continuing  thi;  process  to  22 
operations,  we  have,  10«8»7(h-  =  5.0000(K)+. 

Therefore,  log  5.000000+ =    .698970+. 


402  ELEMENTS   OF   ALGEBRA. 

Example  2.     Find  the  logarithm  of  2. 

log  2  =  log  i^  =  log  10  -  log  5  =  1  -  .698970  =  .301030. 
Examples.     Find  the  logarithm  of  11. 

101  =  10  (1) 

•     108  =  1000  (2) 

Extract  the  square  root  of  (2),  lO^-^  =  31.62277+  (3) 

Multiply  (3)  and  (1)  together,  lO^-s  =  316.2277+. 

Take  the  square  root,  lO^-^s  =  17.78278+  (4) 

Multiply  (4)  and  (1)  together,  102-26  -  177.8278+. 

Take  the  square  root,  10i-i25  =  13.33521+  (5) 

Multiply  (5)  and  (1)  together,  IO2125  _  133.352I+. 

Take  the  square  root,  10i-0625  =  11.54782-  (6) 

Multiply  (6)  and  (1)  together,         102-0625  _  115.4782+. 
Take  the  square  root,  10i-03i25  _  io.74607+  (7) 

Multiply  (7)  and  (6)  together,       102-09375  ^  1 24.09368+. 
Take  the  square  root,  ioi.o46875  ^  1 1.13973+  (8) 

Multiply  (8)  and  (7)  together,      102-078125  ^  1 19.70845+. 
Take  the  square  root,  10i-o390626  _  10.94113+. 

Therefore,  log  10.94113+  =  1.0390625. 

Continuing  the  process,  the  logarithm  of  11  maybe  found  with 
sufficient  accuracy. 

Example  4.     Find  the  logarithm  of  3. 

Take  10^  =  1  and  lO-^  =  3.162277+,  and  proceed  as  before  to  14 
operations,  and  we  have  log  3.0000+  =  .47712+. 

A  table  of  logarithms  to  four  decimal  places  will  serve  for  many 
practical  purposes.  In  the  tables  most  generally  used  by  computers 
they  are  given  to  six  places  of  decimals.  Seven  to  ten  place  loga- 
rithms are  necessary  for  more  accurate  astronomical  and  mathemati- 
cal calculations. 


ANSWERS 


TO    THE 


ELEMENTS    OF    ALGEBRA 


BY 
GEORGE   LILLEY,  Ph.D.,  LL.D. 

EX-PRESIDENT  SOUTH   DAKOTA    AGRICULTURAL  COLLEGE 


'TEACHERS'    EDITION 


SILVER,   BURDETT    &    COMPANY 

New  Yobk  .  .  .  BOSTON  .  .  .  Chicago 

1894 


Copyright,  1893, 
By  Silver,  Burdett  and  Company. 


John  Wilson  and  Son,  Cambridge,  U.  S.  A. 


ANSWERS 


TO    THE 


ELEMENTS   OF    ALGEBRA 


Exercise  1. 

1.  a  plus  100 ;  a  plus  10,  minus  2 ;  etc. 

4.  q  plus  t,  plus  8  multiplied  by  m ;  etc. 

6.  m-\-n-\-r~t]  etc. 

7.  m  -^  n  -\-  b\  m  +  n  —  b. 

10.  X  —  m  ^  n  marbles. 

11.  7i  +  a;  -f  y  4-  6  4-  w. 

12.  a-rrti-\-n  —  k  —  X  —  1/. 

Exercise  2. 

2.  A;  times  ^,  plus  m  times  A;  divided  by  c  times  w,  plus  a 
divided  by  6;  or,  to  the  product  of  k  and  /,  add  the  quotient 
obtained  by  dividing  the  product  of  m  and  k  by  the  product 
of  c  and  n,  and  to  this  result,  add  the  quotient  obtained  by 
dividing  ahy  b;  etc. 

Exercise  3. 

1.  xyz;  5mn;  Sxy-j  15abmn. 

^      ab  ab  25  mn  r      , 

2.  — j-T,    T.  3.  ■ ,  etc. 

a  -\-  b     a  ^  b  m  -{-  n 


ANSWERS  TO  THE 


a      mn 
4.   10.     ^.     -^.     20  amn. 
o  o 

6.    18  mi'.     18  m 71  +  18  m r. 

mn 
8.   aoc  -\-  mnr,  13.   9.     . 

G 

11.  25.     ^.  14.  5.    ^»'  +  ^^' 

m  r 

dt  dt  dt 

15. It. \-  bt.     r . 

n  n  n 


Exercise  4. 

1.  m  fifth  power;  3,  m  fifth  power,  x  second  power; 
etc. ;  a,  h  second  power,  plus  h ;  etc. 

3.  10,  ah  tenth  power;  m  third  power,  n  third  power, 
m  n  third  power,  etc. ;  the  third  power  of  m  second  power 
minus  3  n. 

4.  Etc.;  3,  a  second  power,  b,  times  the  third  power 
of  a  minus  b  second  power ;  a  second  power  plus  b  second 
power,  times  the  second  power  of  a  third  power  minus  b 
third  power. 

6.  10  m,  plus  n  fourth  power,  times  the  fourth  power 
of  10,  n  second  power,  minus  m  fifth  power  is  less  than 
15  a  times  the  second  power  of  x,  minus  y  second  power, 
times  the  third  power  of  x  plus  y ;  etc. 

8.  m  +  n.     2x.     (a  +  by.     (x  —  y)\     5  (x  —  yf. 

9.  (x'  +  yy.  (icH- 7/2)2.   m^x^y^.  ^xy^.   {x'' +  y"")  {x^  -  y% 

11.  1  m'^n^—2n'm^+^a^h'^+^a''h^  +  5a\     '/a-\-b^ 
m  —  n,    .'.  (a  +  by  =  (m  —  ny. 

12.  .'.X  =  m^,'.'x  +  3m^  =  2x  +  2m\     a  +  a  +  a 

to  ?i  —  2  terms  =  (n  —  2)  a;  etc. 


ELEMENTS  OP  ALGEBRA. 


Exercise  5. 


1.  3;  448;  60;  180;  64;  9375;  390,625;  1792;  etc. 

2.  144;  60;  64;  250;  4;  40;  etc. 

3.  T^;  3;  7;  75;  32,400;  288;  etc. 

4.  6;  60;  2;  3;  5^;  72;  0. 

5.  9;  160;  2048;  81;  8| ;  13^;  ^;  H- 

8.  20;  11.  12.   2;  0.  16.   4|§f ;  2^. 

9.  0;  12.  13.   249;  134.  17.   1;  ||. 

10.  14;  6.  14.    22;  ^.  18.   5;  58^. 

11.  18;  -14;  2.       15.    ^;  36.  19.    i;  I4. 

Exercise  6. 

1.  11,  negative.     11,  positive,  etc. 

2.  0  is  6  units  greater  than  —  6 ;  etc. 

3.  6,  3,  9,  12,  11.     10.     b-a. 

8.  2  times  the  expression  in  brackets,  3  i  in  the  posi- 
tive series  plus  5  a  in  the  negative  series,  and  from  this 
result  subtract  6  times  the  expression  in  brackets,  a  in  the 
negative  series  plus  b  in  the  positive  series ;  etc. 

10.  Value,  26. 

11.  Value,  1  +  (-  a:^^)  +  (+  a;«)  +  (-x), 

12.  Value, -2.         19.   80;  -624^^.         26.  56;  -15. 

13.  Value,  0.  20.   3;  15^^;  1^.       27.  2;  15. 

14.  Etc.     14§^§.      21.   9;  2;  15.  28.  102. 

15.  Etc.     17.  22.   127;  21;  6;  1.     29.  52;  -18^^. 

16.  Etc.     6.  23.   3 ;  6.  30.  —  15 ;  12. 

17.  Etc.     ^.  24.   —  4  ;  6.  31.  2  a  +  b. 

18.  Etc.    -2r»^.     25.   16;  55. 


ANSWERS  TO  THE 

32.    5«  +  (6  -  1).  33.    h  +  ^±_^. 

34.  x-\-x-{-x-{-....toa  terms,  or  ax. 

35.  n,    n  -{-  1,    n  -\-  2. 

36.  m,   m  —  1,    m  —  2^    w  —  3,   m  —  4. 

37.  (a  —  1)  m ;    (m  +  w)  m. 

38.  X  —  S,  X  —  2,  X  —  if  X,  X  -\-  1,  X  +  2,  X  -\-  3. 

39.  a- -2/".  ^    ,   a; 

40.  2x  +  '^  +  abc,  41.    ^. 

b  y  —  c 

42.  a"  +  -  —  5  ( ~\  >h  —  x. 

43.  (cc^  —  ^»«  4-  2/'")  ^  ^"  <  g'^". 

a**  «  —  ^> 

44. x""  — 


b""  a'  +  b^ 

(A 

45.  — ^  —  .X  ?/  +  cc  +  X  +  cc  +  •  .  .  •  to  w  terms  —  a"*. 

46.  x"*  H ■=x  —  y-\-  (4a+^»  —  m)  +  —  . 

47.  5^8  — 3  6i^Z>8  + 2^»2.        49.    %x^y'^  —  a^b\ 

48.  3^2- 2cc«?/ +  ^3.  50.    +  6a(ir  4- 2/-^). 

a:  —  y 


Exercise  7. 

1.  (+15.1  a).         5.    (-5£c2).  9.    (+3.8a-^c(/). 

2.  (+15  ax).         6.    (+/2^).  '  10.    -7.71  (^'  +  c). 

3.  (+41c).  7.    (-21a8).  11.    +  5.81  (x  -  ?/)8. 

4.  (+  2  a  ^  c).         8.    (-  3.5  a"  U^).  12.    -  3/^  (|)  . 


ELEMENTS  OF  ALGEBRA. 


Exercise  8. 


1.  (-6.69  a;) +  (+3.5^).         4.    (+2^  x') -\- {+ i  ab). 

2.  (+Ha)  +  (-5'5«^)-  5.    14.9  a- 2.67  (x-y). 


3. 


(+8a'a:)  +  (+7c«x').        ^    6.1Q-2.3Q. 


Exercise  9. 

1.  5  y.         2.   4  //  —  5  a.        3.  x.        ^.  a  +  b  -\-  2  x  -\-  t/. 

5.  —33  ax  —  4:bd  -\-  5ni7i. 

6.  -2a  +  ^  +  c  +  </  +  ^'i  +  a^. 

7.  —  8  a  i  c  -j-  53  a  6  m  —  20  c  m.  8.    4  w. 
9.  3  a  -h  3  />  +  3  c  +  3  rf. 

10.  4.25  a^  4- 3.3  f  +  2  6.  11.    2{^^a+l^^ab. 

12.  iVo  ^'^■^  +  -'^l  ''^  —  2t\  w  n  —  2^. 

13.  —  ac+1.92frf.  14.    —  (1*^.5  a^b+ .25  a  b^-{-b\ 
15.  7.8  (m  —  ny  —  6.03  («  +  y)^-  16.    a^  6»  +  x^y, 

17.  7tJ5  a^  +  5|i  «  a^«  +  T^^  //«  +  3^  x^  U- 

18.  I  «•  4-  J\  Z^''  -  1.3  c»  -  «2^  +  m  a^c-.2ac^-^l}fab^ 

+  ljgga6c  +  2*3c-1.3^c'». 

19.  21.33  a^  +  8.37  ar^  -  5  x  -f  8.5. 

20.  2.6  a*  b^  r»  +  2.8  «« 6»  c^  +  3.91  a^  6»  c*. 

21.  6.09  a*  +  h  f'  +  4.97g  6  c  +  3.5  c"*  —  3.03. 

22.  \aX"l.Sab  +  5.2x  +  ^xy. 

23.  12.91  a  +  4.1  y  —  6.82  z. 

24.  3  o  6  +  10  J-  y  4-  9  a-  //  —  .T«  y  +  22. 

25.  -  J(m-3a-)". 

26.  5  a*.  29.  <?»  +  ^.«  H-  c«  —  3  a  6  c. 

27.  10a«  +  86«+12a  +  12.      30.  0. 

28.  x*  —  y*.  31.  4  a:'"  +  2  a"  —  a". 


ANSWERS  TO  THE 

Exercise  10. 

1.  lOaHc;   -SSab^xi/,  7.  ^  (x  +  y). 

2.  2x'y^;    0.  a  -3ax\ 

3.  —daxy,   xy^  +  bc.  9.  26axy\ 

4.  —3.75x\  10.  %x^y. 

5.  -2.72  a  bc^  11.  8.9i(a  +  ^)2. 

6.  —13.1  VI  np^^  12.  4TV5ra^". 

Exercise  11. 

1.  —2x  +  4:y  —  3z.  5.    ~-2x  +  y  —  :^z. 

2.  -2x-y-4.z.  6.    -^.T  +  |2/  +  i 

3.  —2a  — 10c.  7.   2a^  +  3aH  —  3hc—10, 
^.  Ix+^y  +  l^z.             8.    2x^y  +  2abx  +  l. 

9.  abcxy  —  3aby-}-5bx  —  3. 

10.  2^372/  —  5ac?/  —  2a^c  +  a  +  l. 

11.  .6  ic*  -  1.8  x^  +  7.03  cc  -  9. 

12.  —x^  —  1.2x*  —  2x^  —  2x-\-  2. 

13.  ^^  77Z8  —  I  y«,  7^2  _  J  s  ^2  24.     7^8  —  «. 

15.  ^^m^--J^y  +  2lin-^x. 

16.  2.5«2^,c  — l^a-?/"- 4t^c.  17.    0. 

18.  2  a;*"  +  4  ;?;"  y"*  +  y"^. 

19.  1.8  ^/"^  x^  -  1.7  a  ^»3  .X  +  .37  b""  c  x""  +  .7. 

20.  a^  _,_  3  53  _  g  ^4^         21.   3.55  ic^  +  33. 2y^x  +  S.5x  z\ 
22.  ic'»  +  2y"'.  23.    |  iy  _  1.9§  2/^  +  1-7  ?/«  +  3. 


3.  2-2a^-2a-2. 

4.  2  a  ^  —  ^2  _  ^2^ 

^2  n  yin  rjf.Zn  2  ^*" 


Exercise 

12. 

1. 

—  4  m^  —  2  w  - 

-  5  7z2  +  6. 

2. 

4.x^-Uy. 

5. 

—  2  m. 

6.    4?/ 

w-l 

ELEMENTS  OF  ALGEBRA.  9 

7.    -xhyi-ei/^  +  Sz'  -^1. 

9.    -  ^  ari  +  2.9  a;*  yi  -  33  yl 

10.  3.9  cyi  +  1.6i  ax  +  .31  ^»  -  1.2  7m  -  5^%. 

11.  5.5  6"  —  3.25  «"•  —  4  w.  12.   y-  —  x^  —  1.2  i". 

13.  ^  c'  —  l.Ta'"  —  o^b"*  —  2  d". 

14.  c»  -  i  (a»  -  by  -  12  (x»  +  y2^«. 

15.  2  a^  21.    6  rt-  —  2  a  ^. 

16.  3  a;«*  +  2  x^*  —  6  a;".  22.    aJ  ^i. 

17.  2^2;— I  a^- 2.  23.    4y  a^- ^a»  — 4.5a  +  3f. 

18.  \  //i«  +1.  2^.    llbc -\-l  mn  —  ^xy. 

19.  7  J/''  +  18y  —  4.  25.   4 x*-^  —  2x'''-^. 

20.  0.  26.    x"— 4. 

27.   64  (x  4-  y)*  +  .3  a-  +  x"*  -  3^  a-x"  -  C  +  3. 

Exercise  13. 

1.  7x8;  15a«x«;  2a'*^*x«. 

2.  ix"y»«m««";  2an»c«<;»w»w». 

3.  a6»c»y"^*;  ?a"x»y"«. 

4.  30«"x*y";  ix'"  +  *y"+^ 

5.  2a-^'/>^V^*»x^<>y";  Ti5a'Z»"r"j'"  +  »y"+». 

6.  vSa"+>6''  +  ''x^+V"^*;  ^'/A 

7.  a'"  +  -6»+»';  5.7 x*y".  8.    a»6»x«y«;  a^-ft^". 

10.  2a»x»y".  15.  .765a«mM+'x— "-'1. 

11.  3a*'+''6"    ♦c'+'"rf*.  16.  l|a'"+*'-**^-"//"+*«'". 

12.  2a*  +  »c*+-+*rf«x'"+".  17.  l\(a-\-hy\ 

13.  2  a*l  m^  n**  x*»  /.  18.  3  (c  +  ^)><>  (a  +  ^)". 

14.  5  aHi  a;*A  yi.  19.  ^  («  -|-  ^,)-+ *»  (x  —  y)'"+«. 


10  ANSWERS  TO  THE 


Exercise  14. 


1.  —2S10anG',  a^x^'f.  6.  -3a-H-^c^^d. 

2.  w'hx^y';  —ha^'^hU^d}.  7.  an-^(Px\ 

3.  4:aHcxy)  .Sacm^x^T/^  8.  —  .3  a^^ x^  yi -^   aH  x^. 

4.  ^a^bcdx^yz;  a^^+H'+^x^"" y\ 

5.  —  3  a^b^x^yzvw,  a'«+"  +  i^>"  +  'a:"*  +  ^  ?/"+\ 

9.  —  m^  n  x^t  yH '^  —  2.1  ai  bl      11.  A  X  3"  mx^  y'^^'p  q"". 

10.  a^'^ ''■  a aa--- -to  10 ni3iGtovs.  12.  _  2^«  +  »"3^°«  +  ^"  +  i. 

Exercise  15. 

1.  abU^+aHc^-aH'c,  ^  aH'*  -  :^  a^^^  c^  -  ^  aH^^  e^ 

2.  5  a^  Z*»  c^o  —  a^  b''  c""  —  2a^b^  &'' ;    .12  ic^  y^  -  .1  £c^  / 

3.  3m^  n  —  1\  w?  ii^  +  3  m  ?i^ ;  9?- y  —  xy'^  —l\x^  y^. 

5.  9  a«Z»ar3— |«H^aji3^  .3a^>«ar«;  p^  x'^'^r  —  p  qrx'^-^'' 

—  pr^  ajtn 

6.  3«'"^>2_2«^«  +  4  rt»»  +  l^n  +  2.      .Ga^'^Z;^^  -  2  6/"'-^^>2p 

8.    a^i  ^^s  -  a^^  ^»^^  -  a^  b^^  +  a^b;    x^  y-h  -  x^  y\ -\- x^  y 

—  .Bx^yl 

9.-2  .T.3  ?/^  +  8  a^2y6  _  8  _^  yv .  ^„2  ,^o  _  2  ^^p ^2?  ^  ^o  ^^2,^ 

10.  ^V  «^  ^'^  ^^  +  ^7  «*  ^  cc'  +  ^  a^  Z*^  ic^ 

11.  —  cc§  ?/  +  .-^^  ?/^ ;  ai  ^'«^  —  6?i  b'^\ 

12.  45a!i3i7/°i-45£c'«^/3>. 

13.  ^V  ^'^^''  —  :\  ^'''^"■•'^'^'  —  -2  ^>'3\tV  4-  ^  //7-5. 

14.  180  a-Sm/.-am  _  189  ^-4m  ^--im  _  |4Q  ^-2m  Jl-2m^ 


ELEMENTS  OF  ALGEBRA.  11 

Exercise  16. 

1.  a*-{-aH^+b*;   a*  +  4  a^x^  +  16  x*. 

2.  x^  -\-  i/;   ^xij. 

3.  2/'  +  -'y'~lly'-12y  +  27;   y*  +  a  y»  +  « V 

—  a  t/  z  —  a*z  —  a*. 

4.  \x*-l,^^x'  +  ^jf',   .64  a»  4- 3.24  a  ^2  -  2.7  ^»». 

5.  a^-{-2x'-2x'i/  +  x^-xi/- 7/ -{•!/', 

l^x*-  1^  ax«  +  i  a^x^  —  I  a\ 

6.  a;«  + 8a:y  +  y  —  1;    H  a:^  +  6  aic*  +  f  a*x«  +  faa^* 

+  3a^x^  -{-  la^x  _  ^  a«x«  —  ^  a«a- -  j^jj  a». 

7.  a-'  -  7  a;«  +  21  a;*  -  17  a-*  -  25  a-«  4-  6  ^2  -  2  a;  -  4. 

8.  a»  +  2^i«4-5a*  +  2a24.  1. 

9.  x»  -h  1.25a;'  +  .25a;«  +  Sx*^  +  .5.r*  +  .25  x'  +  1.25  a:^  +  1. 

10.  4  +  32  a  -  4  a^  -f  25  a»  -  6  a*  4-  «». 

11.  x»  —  32  y*.  13.    a«  —  3  a  ^>  c  +  //  4-  c». 

12.  x^*  4-  y^^  \   x^  -\-  i/.  14.    «»  4-  3  rt  ^»  c  4-  ^'  —  c*. 
15.  a*^»2  -  rt^-2  4-  ^*<^  4-  ^f/^.  16.    —  8  a-2 //. 

17.  .3  m«  4-  2.90  «  w  —  3.01  i  m  -  .3  71-^  4-  3.01  a  7i  -  2.99  b  n 

—  .01  a^  4-  .1  ^. 

18.  a^x^'*  +  b^x'^'*  +  1^  -\-  2a^»a;'"  +  "  +  2ara:"  +  26rx"; 

ari  —  yl 

19.  a*"  +  2a'" 6- 4-  b^"',  a*"*  — i*-;  a:l  4-  &^*+  J5a-*4-  ^'. 

20.  6a;'"  — 4a;;/'-*~9ar— V*  + 6y;    a^a-m+a  _^  ^^^,3.n+2 

+  a^6x*  — a^ar'"+'  — i'^x-^'^-ff^^'^a:*  — ax""  — ^x" 

—  a  Ax. 

21.  3a*'"'x4-3a2  +  -y  +  f(2''^'"  — Sri^'^+^a;  — 3a'-*-''?/  — a'" 

4-3a«-x2  4-3a«xy  4-  «*"^;  x5  — x=//-?  — xiy  +  y*. 

22.  .04a  — .09  61;  x  —  y.      24.    l  +  x  +  x^  +  x^  +  x^  —  x^ 

23.  X*  —  i/-\  25.    X*  —  5  a*  x*  4-  4  a*. 


12  ANSWERS  TO  THE 

26.  120  ic*  -  346  x^  -  205  x^  +  146  a;  -  120. 

27.  x^  +  x^  +  1.  28.    x^  +  a*x^+a\         30.    a^"*— Z.6«, 
29.   3a^-oa'b-SaH''-j-7  aH^  +  6aH^-2ah^-  h\ 

Exercise  17. 

1.  a2^2a-15;  &2_^^>-30;  a;2^7a;  +  12;  a;2-3a;-4 

a;2  -  5  £c  -  14. 

2.  ^2— 14ic  +  48;  a2+4a-45;  a2_4^_32.  4a;2-18a;+20 

9ic''^+ 6aj  — 35. 

3.  :c6  —  7ic^2/^  +  12?/^;  a;^  +  xy  —  ^y'^;  a^"^  +  a""  —  2 

9:i;i«-27a;5  +  20. 

4.  4a*/-8ay-32;  9  a^ a;2_^  9 a  ic - 28  ;  a;«  -  a a^^-12a2 

•  x'o  _  ^2  ^5  _  6  ^-2 

5.  4a;2_2(ia;— 2^2;  4x2"  +  4  a  x*^  — 15  a^;  9^:2  —  60:3/ 

—  2  y^;   Ax^  —  Amx^  —  24:  rn}. 

6.  a;2_6^^_|_5^2.  ^2^3  ^  ^_4q^,2.  ei^  _- 8  a^  £c  +  12  a;^  5 

25x2»-5a2;K^o_i2a^ 

7.  25  a:^«  -  25  cc5^2  ^  6  2/^    9  a^«  +  3  a^  (2  a  ft  -  4  a  ^/) 

-%a'b';  «2"  +  a"  (3  -  6)  -  3  Z». 

8.  16^2  + 4a(^>- c) -^»c;  2na'  —  10a(b^c)  +  4.hc', 

a^y'^  +  I  aa;^/  +  ii^^'-,  «^  +  i  «' —  I- 

9.  4ar  +  26 a;^  +  12;  4  a  +  2  a^  (5  —  3  a  a;)  —  3a  ^»a;; 

.09  a;  2/2"  -  .3  a;^  ?/"  (a;  +  2/)  +  «^  V- 

Exercise  18. 

1.  4a;^-92/^  a:^  -  4  2/^  25- 9x^  25  a:'^  -  121. 

2.  4  a;2  -  1 ;  4  a;2  -  25  ;  25  x^y'  -  9 ;  c^  -  a^. 

3.  c*  —  a^\  m^ri^  —  1 ;  a^  y^  —  }p-\  a^  x^  —  1. 

4.  x^  —  y^;  1—p^q^;  m^  —  n^\  a^"*  —  a^". 

5.  25x2?/-2-16  2/^  25a!^-9y^  x^-9x\ 


ELEMENTS  OF  ALGEBRA.  13 

6.  ^a'^x'  —  b'^y^',  m-^^  —  ii-^^-,  100  a"  2"' _  169  6"''*. 

7.  m  +  w;  16a  — 400a;*;  al  —  b-%, 

8.  121  a:- 900  y;   225  a*  *•- 256  a  61. 

9.  \aH-^-tb-^x-^;  a^-b\ 

10.  aH^-l\  16  a*'"  —  256 a*". 

11.  625  a»2  -  1296  b^ ;  a"  ^^  _  «« 6". 

12.  x-\-x-'y^;  |ec-'"-^af^"*- 

Exercise  19. 

1.  4  a**  —  a-"  +  6  a"  —  9 ;  20  a^  —  3  x2«  —  a;'-  —  6  x^**. 

2.  4  a*'  4-  li  a"  +  3§  a^^  +  6  a'  -  15  ;  2  «  +  ^5  a;5  -  |  a;4 

3.  _  §  a»  +  6^  at  —  §  a2  —  4^  al  +  6  a*  —  a«  +  I  a-l  -  a-2 ; 

6  a*"—'*'  +  3  ar--^b^f  —  2  a^-sp^-sp  _  ^p  ^-p 

4.  25  a^V*  +  20  a;2«2/26  _  15  a:«  y*  _  12  x«  y%     a^*  +  a" 

+  a—  4-  a-*". 

5.  .09a«-.156a*6  +  .22a*6«-.488a»6«-1.39a«6*+.3a6* 

+  .rib\ 

6.  1  _2a;J-3a;i  +  2aji  +  2a;i;  a\  —  al  ^-  4|al-8ia-i 

+  a-i  -  2  a-i  +  i  a-»  -  i  a'^  +  «-«  -  2  a"*. 

7.  -2a;t  +  2x5-3x-5a;-l  +  a;-84- 3a;-«;  al"+a-i". 

8.  x""+'  — 2a:*"+'*'  — 2ar2''+2  4.2a^''  +  '  +  a:''^*  +  2  3f +  '— a^+* 

-  2x*  +  x"-';  j;2n+2_4^2n  _^  12a:2--i  —  9x''*-*. 

9.  2a:»"-*-4a;"'-'  +  2.1ar«»-.9ar*''-'+.lx*"+2+.2a;'»+* 

4.  2a;»"-»  — Sar*— 2  +  x«— »  _  ar^-. 

10.  9af+"-'  — 34a^+— »  4-  29  ar  +  -  +  »  —  «•"+"+*. 

11.  2a;^+>  —  63:*+*  4-  2x«+'^  —  4a:'+*  +  3a:*«+»  -  9  a:»«+* 

+  3a:*^+»  —  ^Q^'^''-^  —  4a:»'^+»  +  12x«'+*  -  4  a;'«+» 
+  8a^'+^ 

12.  ITar^-^V*"^*— lOx^^+V  +  l^x^'-^V"^'— ^^"""V*"^* 


14  ANSWERS  TO  THE 

13.    m^  +  '-— 3m^+'-i  7i+  m^+'-^Ti^—  m'+'-^Ti^  _  3^p+r+i  ^ 
+  9 mP  +  '- n^  —  3  mP+'-'^ n^  +  3mP  +  '■  -'^71^  -{-  7iiP-^'-+''  71^ 

15.  2  x^y  —  18  x-^^^  +  6  a;3y  _  18  a:'^  /  +  4  a:?/^ 

16.  f  —  x-^"*;    ^^x'-y-'^b  _^^^-'zay2i,^ 

17.  x^"—  2/2m.     ^_^^_20;     49x2  —  9^-2, 

18.  16  x2  —  8  x-0  -  15  a:-2 ;  ^cH-^  —  ^\  ^»  Z/"* ;    a^n  _^  ^14  ^n 

+  49-9a-2«. 

Exercise  20. 

1.  16aH^',  21a>^m',   32x^y^',  .00032  a^o^^i^c'-O;  .01a2»^,2«. 

2.  49a«^^    121  a2&4c«<^^;    -  27  c^  a:«  5/I2  ^i^ .    27a^^^»«y^ 

25  a^^^^^^yo  ^20^ 

3.  256  a«  6«  ci«  ^^^4  ^^32.     ^10  ^10  ^10  ^10  ^10.     _  ^^9  ^e  ^s .     ^8  js. 

729a^6^8c^;    -^aH\ 

^4nj.3n^«^2n^ 

5.  — 8a-^'7»'«";  m"=;  ««';  a^b-'^c-^^',  m""*'/!-"*'";  -8;  a^n^  1. 

6.  16a*5-«c*?^^;  ri^^  +  i^smn^  ^»»«  +  ia-m.  ^rri'n^x^^f. 

7.  -54am3  7i»a:^2.  81  a^« 5«^ c^* m^^ ;  81  a^^,-*.  a«^-«  +  i. 

-243a^-20c-'#ic*?/§. 
9.    _ ^14^21  ^56^28^.  _a^rm,/»5  ^-'"'b^'k^*;  _8a2i^,i4.  x-^y^\ 

10.  X^2«^42m.    ^21^65.    4"  ^7n  J7n  ^7»«  .     _^7m»n«.     J^Uk  ^Ukm^ 

11.  m"  (a  —  3  <^)P"  (x  —  7/)'"  ;    (a  —  3  c^)i"'  (a;  —  2/)^" ; 

3"  (a  —  &  +  c  +  c?)"  (a  —  cc)""*. 

12.  a" ^>" c"  (a  —  &)"*"(cc  +  y  +  z'^Y" ;  ct"'  (x  —  y  +  ^)^"'  (cc — y"»)8'»*. 


ELEMENTS  OF  ALGEBRA.  16 


Exercise  21. 


1.  a;«  +  4  X  +  4  ;     m^  -\-  10  m  +  25  ;     n^  +  14  w  +  49  ; 

a«-20tt  +  100;  4a;*  +  12a;y  +  9y^  a"" -\- 6ab  +  9b^; 

2.  J-2+  10xy4-25y2;  9x«-30a:?/  +  25  2/'^;  4^2 +  4^26 

+  a-62;  25x2-30x^^  +  9x2^2;  25a2^,2c'2— lOaic^  +  c*; 
a;iy-i_4icy  +  4?/*;  a^m  ^  Qa'^b-"  -{-  ^b-'^\ 

3.  4  x^  4-  12  a*  X  +  9  a  ;     x^  f/^  +  2  x^  y  -{-  x-t -,     9  a-* 

+  30a»6-»  +  25ai»^»-«;  l-2x  +  x2.  l_2cy+cV; 
m2-2w  +  l;  a2^,4_2a^-^+l;  ||  a^-- /^  a*  +  ^i^. 

4.  ^a2^-*-f  3a6-»x-»  +  |6-2x-2;  j^*'*^*'-  — 2j9*^'"r«4- r^'; 

.00000004x2'"  +  .000002  x*"//"  +  .000025  y2». 

5.  3»j/wMtV  —  2w»7i''  +  'y/  +  Y^^''^^"';    icV  +  2y2a;« 

+  //2^2_^3.a-g2  _j_  2ar2y^ -I-  2xy«2.    4x*  +  5x«+l 

+  12x»-6x;  x*  +  6x2+  l_4x»  — 4x;  x*-4x2 
+  16  +  4  x»  -  16  X. 

6.  4a?*H-13x«  +  9-4x»-6x;     x«  +  25x*  +  4  -  lOx* 

—  4  X*  —  20  X  ;     16  w*  4-  w*  ^2  +  7i»  4-  8  m^  n*  —  Sn* 

—  2m^n^\  x»  +  9x2  +  4__6x«  +  4x*  — 12x. 

7.  x2  y2  _|_  4  ^8  ^  1  _  4  ,j  jp  y  _l_  2  a;  y  —  4  ?i ;    rti^  ■\-  v}  ■\-  j^ 

-{-  q^  —  2 m7i  —  2mp  —  2mq-\-  2np  -\-  2nq  -\-  2pq] 
a;«  + 8x^  +  16x2+  9-4x'^-14x^-12x;  1  +  3x2 
+  3x*  +  x«  +  2x  +  4x«  +  2x»;  x2  +  9^/2+  4^2 +  ^2 
+  6xy  +  4ax  —  2/>x  +  12  ay  —  6  bi/  —  4ab. 

8.  4x*+5x*-  17x2  +  9  + 18x»-6x;    x2+4y2^9^2 

+  4n2  — 4xy— 6x2  +  ^  n  x  -\- 12  yz  -\-  ^  n y  —  12  nz\ 
^2"  +  n2"*  +  />2»  ^  ,^2«  ^  2  ?;^'•7^"•  +  2??^''/>"  —  2m* q"^ 
+  2n'-/)»  — 2n-^'"  — 2/>»^'";   ^a^  +  g^  +  f  — 2a6 

—  a  +  9  ft. 

9.  4a«  + 4  62+TVc2~2ai  + iac-Ac;    x2"  +  y2m  _,.  ^  ^2 

+  i  62 _  2  X*  2r  +  <»  X- -  2  6x*  —  ay»  +  ^  *2r  —  J\  aft; 
^.r*  +  3x2+ |-$x«"-3x;  |a:«+}x«+^-2x» 
+  ^x*-§x». 


16  ANSWERS  TO  THE 

10.  1-f-  ia:+^V^';  i^-'-^^'  +  ^V-i^'  +  Aa^;  ^^a'^ 

4  a;*3  +  25  x  +  49  +  20  a;i  +  28  ici  +  70  o^i 

11.  9x  +  4.x^  +  ^xi-{-x-i~12x^  +  2a;i  — 6a:J  — |£ci5  4-4ccs 

5-23  x3i 

Exercise  22. 

1.  a'-7  a«^»  +  21  a^^2  _  35  ^4^8  ^  35^8^4  _  21  ^2  ^5  ^  7^^.! 

— ^>';  a^  +  6  a°  ic  +  15  a^x^  +  20  a'^c^  +  15  a'ic*  +  6  aa:^ 
+  a:«;  a^-4:a'c+  6  a^  c'' -  4:  a^  c^  +  aU^;  a^-16a« 
+  96a«  -  256a»  +  256;  16  +  32a  +  24.a''  +  Sa^  +  a^ 
a^  -  5  a*  +  10  a^  _  10  ^2  _^  5  a  -  1 ;  1  -  5  a  +  10  a^ 
~10a»  +  5a^-a^ ;  16  a*- 96  a^i  +  216 a^b^  -  216  ab^ 
+  Slb\ 

2.  ic2-12ici+ 54ic-108xi+81;  a^a:^ -15a4a;«  +  90  a^a;' 

-  270  a'x^  +  405  ax^  -  243a;^«;  x'  -  15  x' +  90ic« 
-270a;2  +  405ic-243;  Sa^x^-ma''Px''i/-^54:aH'xy'' 

-  27  ^«  2/«;  16  a*  o:^  +  96  a^  ^  a;^  y  +  216  a^  V  x"  y'' 
4-  216ab^xy^  ■\-  SI  b'y\ 

3.  a«  4-  12  a^  +  60  a"  +  160  r^^  +  240  a^  +  192  a  +  64; 

a«  -  12  a^  +  60  a'  —  160  a^  +  240  w"  —  192  a  +  64; 
16  —  3^  a  4_  I  ^2  _  ^a^  ^8  _^  ^  ^4 .    ^1^  ^4  _  3  ^3  ^ 

+  V-  <*''*'  -  54.ab^  +  81  ^4 .     _i^  «4  _^  ^  ^3  ^  _j.  ^  ^2  ^2 

-  3^  a  6»  +  ^V  ^^  ai«  +  10  «» ^  +  45  V  ^2  +  120  aH^ 
+  210  a^  b^  4-  252  a^  b^  +  210  a*  6«  +  120  a^  b'  +  45  aH^ 
+  10aP  +  b^\ 

4.  a  +  4  4- a-i  — 4a^--2a«+4a-i;  4x^+4^4-8:^2^2. 

l  +  3a2_^^4_,_2a4.2a^3;  _l_5a2-4a*  +  2a 
+  4a». 

5.  16a8i  +  64«63;  _7  +  20a-16ct2^4a«;  1  +  2  •  3^  •  5J 

^  2  .  2i  •  3^, 


ELEMENTS  OF  ALGEBRA,  17 


Exercise  23. 


1.  3a^bj  6ab\  ^abc-^\  f  mA. 

2.  w«;  a^  a-H-^c^-''',  a?";  2^'. 

3.  5al  AJa;-^  ^a-i**;  3a••-^mx'-^ 

5.  (5c  — y)2;  (a-c)*;  |  6*-i  r"' A;«-«. 

6.  Oa^^c-^  a'"'-"';  2'm'w"^ 

Exercise  24. 

1.  2a;;  -2a*6*c«;  -oa*;  1. 

2.  —3a;  -^a^i^c*;  .Saft-^c"^;  -SOa'^ir*. 

»    a    A         5  m~  *  a;*  V*     n  i  ^       2    s 

*•  —  ^^  «7wy«;  —  31  ma: 2/^"-*"'. 

5.  -6a:"-'/'';  -  ii(a-^)c"^   ^a-i^^-i* 

6.  10  a;l  y-  i  (a;i  — -  y')  A ;  |  w  w  x  A  y. 

7.  -  4  a*  (x  -  y)^  2- 1 ;  m"  n""  (a;  -  y)^''  (y  -  «)H. 

8.  6a-^6c;  -2a'»  +  "6-  +  "c-'. 

9.  -2a-Sil. 

k  13.    -^a^-^-d^-^-c*-"". 

11.   2a6*c*rf°x-».  14.    Ha"'6''-ic-Ix-». 

Exercise  25. 

1.  1  +  3  ay  —  4  aV;  3  m^  /i^  -  m  7i  -  2  +  -?-  . 

mn 

2.  1 -J«c- aft  r2  + aH*c^  6a;*— V- a:  4- 40. 

3.  4a«+  f  a-3  +  -;  4a6-6-^-a-^ 

a 

4.  9aftr«  — 12aft'  — 5c*+  ^aftc. 

2 


18  ANSWERS  TO  THE 

5.  3mn  —  rn^n"^  —  3  n  -{-  5  th  n^  -,  x"^  —  x^^  ?/'.  - 

6.  2  a  —  3  ^>  +  4  c;  —  J3O-  ^^'^  +  2  n". 

7.  _  3  ^4^7  _  I  ^2^-1  ^  2  a-i;  .9  7i^e^  -  1.2  ?iA. 

8.  36f  05  2/"  + 10-^-8^;  3mi«  + 160m-80  7y^-l7^. 

9.  ^m-2  _  2  a-«>-2  4.  3  a!'-^\  m"-2  —  m**"^  +  ??*"  —  w" '  * 

10.  —  3  xy""-''  —  2d^x^  +  4.a'^x  if''. 

11.  _  a»»-i  y^  4.  a"*  ^>  —  a'*-^;  —  f  a^  ic'^  +  \l  a  x^. 

12.  6  a  —  ^  ^  —  6';  xi  —  3  ars". 

13.  3  m5  —  f  ?j3  +  ^  r^if  ?ii 

14.  4  (x  -  ^)*  -  3  (x  -  y)2  +  2. 

15.  2«'-i  +  Sx"'?/"  — 18a:'"y-'*^'+\ 

16.  (cc  —  ?/)''-'*  —  m«-". 

17.  (x  +  yf-'  (x  —  yy-"  +  (X  +  yy-'(x  —  2/)«-^ 

18.  3  7?i  —  2  71  —  4 ;  «i  —  a*^  ^>2  +  ai 

Exercise  26. 

1.  Tx''  +  5xy  +  2y^ 

2.  x^  —  x"-^  ?/  +  £C  z/'-^ ;  a^  +  or  ^  —  b'^. 

3.  vy«  _  2  2/2  +  7/  +  1;  7/5  +  2/*  +  2/'  +  2/'  +  .V  +  1- 

4.  JC+22/-.^;    a7  +  a6^_^^5^2_|_^4^34^3^44^2^5_^^56  4.J7^ 

5.  2  a;  +  3  ^ ;  a^ 

6.  .25  x2  —  3  X  ?/  +  9  7/2. 

7.  27x2  +  12x7/+  6  7/2-1;  x^  —  x^ 7/  +  xif  —  y^. 

8.  1  —  7/  —  X  +  XT/;  2  7/3  —  3  7/2  +  2  7/. 

9.  x2  —  X  7/  +  X  ;2  +  7/2  +  7/  ;>;  +  ^2  .    ^.3  _|_  ^^  yj  4.  /jjj  2/i  +  7/i 

10.  X  7/  +  7/  ^  —  X  ^  ;    X'^"  +  X^  2/^  +  aJ5  7/t  +  X?  7/5  +  7/t. 

11.  4x2  —  6X7/  —  8  7/2.  ,      ■    • 

12.  X2  —  X7/  +   X   +   7/2   +   7/+   1;     ^X2-^X   +   yV- 


ELEMENTS  OF  ALGEBRA.  19 

13.  6x*y«-4ar^y«  +  y^  x! +  xyi +  xiy +  yl. 

14.  aH  —  ab^j  x»  +  x-\ 

15.  X-  +  a;  2/  +  y* ;  «  +  "i  Oi  +  0. 

16.  a^- ia+  2;  2*5^*-3«'y+  ixy\ 

17.  a  —  aJ ;  x*  +  x*y  +  x'^y"  +  x  ly'  +  y^ 

18.  X*  +  y-^  +  ;g'^  +  -  xy  -  X  «  —  y  « ;  x'^  +  .75. 

19.  x4-2y  +  z.  22.    2a^  —  3ab  +  ^b\ 

20.  |x*-ia;-§.  23.   6aj-^y-i. 

21.  X*  —  x''  y  +  X y*  —  y** ;  x'—  x*  y  +  x^  y*  —  x'  y*  +  xy*  —  y'. 

24.  a»-2tt«i'-  +  3a*^»^-2a-'^°  +  ^«;  a»+ 2a^i  +  2  a6^  +  ^»». 

25.  2x**  — 4x-y"  + 22^";  x^'^-x'-y"  ^y'". 

26.  X*  —  y"  +  z' ;  3'  +  2'. 

27.  |a^  +  ix»y4-  ^%  x'y^  +  ^h^y'i  x-i'»  +  2y-i". 

28.  yx"  +  zx"  +  r. 

29.  x-i  +  y-*;  x^  — 2Jxy  +  y'*. 

30.  x-J-y-l  +  sr-i;  x»  +  x''y  +  xy»+  y'-    ^^' 


X  —  y 

Exercise  27. 

1.  7/1*  +  //?  n  -f  J/'* ;  a*  m*  -{-  a^  bm^n-\-  a^  b^  m^  n^  -{■  ab^mn* 

-^  b*n*;  m*  n*  +  w^  n^  +  m^  n^  ■\-  mn -\-  \. 

2.  1  +  m  n  X  +  m'  n*  X*  +  7w*  7i*  x*  +  w*  ;i*  x*  +  m*  7i*  x* ; 

x«»y«  4-  xV«  +  x«y 2*  4-  x\//2*  +  xV^*  +  a;«yz» 
4-  x*2%  1  +  ahX'\-a''b'^2?  +  a''Vv+  a*6*x*  +  a'*6«x'» 
+  rrV/«x«. 

3.  a«+rt»^  +  ft*;  x"  +  x'y'  +  y";  x"  + x^V  +  x»y<  +  x«y« 

4.  a"  +  a»i«  +  a*i"+  aH^.+  i^;  x""H- x"- 7/- +  .r'*"?/"'- 

+  2*-x*-  +  2*-x-"'  +  x*-. 


20  ANSWERS  TO   THE 

5.  16  a^ +  12  a^n^ +9  n^',  4:x^y^*'+ I. 

6.  a'^x^P  +  a*b^x^pf"'  +  aH^x'^y^"'  +  b^y^'"',  16x^+2Ax^y^ 

+  36x'y'  +  54.x'2/  +  81^1 

7.  x-s  +  ic-K?/-^  +  x-^^y-i  +  y-i\  xi  +  ic-^/^  +  »^^y  +  ^V^ 

+  x^iy^  +  y}',  a^ x?+  ai b-^x^ y^s  +  a^ b-^xyi+ai  bx^yi  +  b^y^. 

Exercise  28. 

1.  12ba^x^— l^a^mnx'^  + ^5  am^ 71^ x— 21  m^ 71^',  x^-x^b^ 

+  x'b^  —  b^;  x''  —  x^  +  x^  —  x^  +  x^—  x^  +  X  —  1. 

2.  £c5  -  2/?;  64  a;^  -  160  x^  +  400ic  -  1000;  x^""  -  7/\ 

-  1^5  2/^ ;  a^-aH  +  a'  h'  -- aH'' ■\-  cv'  b^  -  a''  b'>  +  «'  ^' 

-  a^  ^7  +  «,  ^8  _  ^9^ 

4.  243  ai<>  - 162  a^  b^  +  108  a«  b^-12  a^Z/^  48  a2^»i2_  32Z;i5; 

ax"  —  5'-^  2/2"*. 

5.  a-^  —  a-^x-^  +  a- 5 ic- «  —  X- ^ ;     a^ x~ ^  —  a^ ^s x-%~i 

-\-  ab^ X- -^ y-^  —  b^ y- 1 

6.  icl«  —  £c"2/T2'"  +  xi'*?/i'"  —  x^"2/i'"'  +  xi"?/3'"  —  ?/?5"'; 

Exercise  29. 

+  ^-22/-2_.^-iy-3+  y-4.  2o6x'-192x'y  +  lUx'y' 

-10Sx7f  +  Sly\ 

2.  64xi«- 96x1^7/2+  144xi2y4_2i6x« 2/'+  324x«2/'-486xy^ 

+  729//2;  Slx^^-5ix^7f  +  36x'y'-24.x^y'-{-16y\ 

3.  Xl2n   _^iou  ySm  _|_  ^8«y6m_  ^6«   ^^9m  _j_  ^4«  ^12m  _  ^2n  ^15m 

_|_  yl8  m  .      7^18  a_jA6a  ^^5  n  _|_  /^12  «  ^^^10  ri  _  ;^9  a  ^^^15  n  ^   ^6  a^,^20  » 

-  A;8«  ^25"  +  m»^" ;     a^^  -  a^  ^'  +  a^  b^  —  aH^  +  a^ b^ 

-  aH^  +  aH^  -aH'  +  aH^  -  ab^  +  b^"". 

4.  7^1  ^1  _  mi  71^  x^  2/1  +  m§  ?i?  £C^  y^  —  7?^^  ti^  x^  ?/5  +  x^  y^ ; 

x^  —  x^y^-^xy^-x'^y^  +  y^',  x-'^-x'^t/-^  +  x-^  y-"" 

-  x-iij-^  +  x-Uj-^  -  x-'^y-''  +  y-^. 


ELEMENTS  OF  ALGEBRA.  21 

5.  af  X*  —  a»  6A  xi  y»  +  a^  b^i  x^  y^  —  a^  b^  xl  if  +  a?  6i?  x  y> 

6.  a'^(iH'-\-b^;  a;*-a:Hl;  a:«-a;*  +  l;  ai8-a«6»4-<^". 

7.  a;»-x«/  +  «*y*-^'/  +  /;  a;''-xy  +  y°;  16-4x^0;*. 

8.  16a;*-36a;«/  +  81y^  a;*-^  ««  +  tV  ^'- b^  a;'+ ^i^; 

9.  ,,12 _  ,,«  //  _|_  /,ii  5  «i« _  ai4  ^2  4_  ai-2  ^4_  ,jio  ^e  _|.  ^8  6«  -  a«  ^»^« 

4-  '*'  b'^  -  a^  b'*  -h  b'' ;  ,V  a^'  -  3^  ^'  f  +  iV  y'- 

10.  >(•'*  —  a» '  ^^•-  +  b''* ;  a«-^  —  rt^s  ^,4  ^  a'"  ^»8  —  a*'  ^"  +  a^«  ^^^ 

-  a"  &»  +  «« 6'^*  -  «*  />-^8  +  6«-^ ;  81  x*  -  9  a!^  +  1. 

11.  .r"'  -  ./•'»  /  +  x""*  i/'-'  -  x"  i/^  +  ^12  2/24  _  a.6  2^80  ^  yj6 .  ^m 

^12  _  an»  +  «•  ^'  -  <^'  b^  +  ^''. 

12.  .r3«  -  x"  //"  +  y»« ;  x«  -  x^^/  +  x««  //^^  _  ^so  ^is  ^  ^24  ^84 

-  x"  1/^  +  x^-  //«  -  x«  //«  4-  y/*8 ;   yes  ;   a«  _  ««  ^4  _^  ^,8 . 

a4  ^«    _    ,^2  /,8  ^9  ,^12    ^    ,^^18  ^^24  .        ^12^^20  _    ^9  ^16  ^,^6  ^2 

+  a«  6»«  m"  /i^  -  «»  6«  wi"  7i«  +  m"  n*. 

13.  2  +  ./,  4-2a  +  a2;  a»  +  i',  </«-^»«;  2  -  x,  4  +  2x+x'^; 

x«+9,  x«-9;  a*^  — 6*,  a^«  +  ^/6^;<+ ^^8;  9««^-4^»^ 
9«6_4^4.    a2_|_25,  aa-25;    a»-J»,  ««+a«i«  +  i«. 

14.  x*-y»,  x^+x^V+^V+aJ^iy^+.v";  ^w+x,  7«*-7//«x 

+  tTi'^x^—mx*-\-x*',  x«  +  .y«,  x^  —  i/;  x*  +  1,  x«  —  1  : 
a-«  +  ^-*,  a-«  — &-•;  a*x*'  +  b*y*',  a»x»' —  (^«  ?/''; 
a;  +  2,x*-2x«  +  4x«-8x+16;  4a«  +  9,  4a''-*9. 

15.  2a  — 6,  16a*  +  8rtV;  +  4an«4- 2«V>8  4-i*;  9a<  — 4/>, 

9a*  4-  4ft  ;  1  -  y,  1  +  y  +  y*  +  f-\-  y*  +  ?/  4-  ?/; 
oa;  +  10,  a«x«-10ax4.  100;  a^x'^-l,  a^x'' 4  1; 
a  +  7WX.  rf^  —  r/'/nx  4-  a^  m*  x!^  --  am*x^  +  m*x*j 
X  y  —  9  ^,  X  y  +  9  a. 


22  ANSWERS  TO  THE 

16.  2  a^  _-  3  h%  16  a'  +  24  a^b^  +  36  aH^  +  54  aH^  +  81  b^""; 

+  b^"";  a^x^"  +  b^i/^"\  a^x^""  —  a^  b'^x^^'y^'''  +  b^y^""-, 
cxP  +  b  y%  c^  x^P  —  c^  b  x^^  y""  +  c^  b^  x^^  y^ »  —  c^  b^  x^^y^  " 
+  etc.  * 

17.  a;-2"  +  2^-2«,  x-'*"  — a;-2"2/-2"+2/~^";  2a;y+9,  ^x^y"^ 

—  36  a;  2/  +  81 ;  a^  f»  —  b^  /",  a^  2/«^  +  ^;«  2/«« ;  c' x'' 
+  ^>*  ^Z"",  c* x^^  —  b^y^""',  XT  —  36,  a;-^  +  36 ;  a^"  —  ^2«^ 

18.  2  a;8  +  3  3/'^  64  a;i«  -  96  cc^^  ^/^  +  144  cc^^^/^  -  216  x^  y^ 

+  324  x^y""  -  486  a^^  y^""  +  729  ?/" ;  16  a;«  +  9  2/^,  16  ic« 

—  92/^;  a'"6"  +  ic^/,  a^"'¥''  —  d^'''h^''x''y'-^a^'^b'^''x^'y'^' 

—  or  b"  x^'' y^' 4-  x^^y*' ;  l  +  2x\\—2  x^  +  4 ic^  —  8 ic« 
+  16  a;«  —  32  a;i<^  +  64  x^^;  a"*"  —  6«",  a"*"  +  ^«»j 
x-^y-^  +  Ij  x-^y-i  —  1. 

19.  ic^r.  ^1  _  ^^1,  y-j^  ^2„  ^-1  _l_  ^^^-i.  a-^jc-i+ 1,  a-2^-2 

—  a-^ic-i  +  a-^x-^  — a-2ic-4  +  l;    fa^ici^H- |^'~^2/"^ 

—  i  2/-^    tV  ^-^"   +   ?^T  ^-"2/-"'   +   T^o  x~l^y-^^ 

20.  ^'s  ci«  +  .3  £ci  ?r  ^  ^^  ^i''  —  3  ^>  cf « ic^  ?/- ^  +  .09  b^  c?« a;?  2/~^ 

-.027  ^>i  d«a;?  ?r  ^+. 0081  a^t  2^1;  ^16a^ic-f  "  +  .090!-^"', 
16a5a;-l«-.09a;-J'";  2-t«aTV-3^  i-^,2-§"«V<.  +  3^^»-i 

Exercise  30. 

1.   3  a^h-^  —  ^^-  a  +  ^V"  ^;  ^~^  —  ^^y~^  +  .V"^- 

3.  l  —  2a  —  2aM',xl-\-xh y\  J^  x^Tjh  ■\-  ?/l. 

4.  {a  —  b  —  c)"^  —  (a  —  b  —  c)-2'«  —  (a  —  b  —  c)"". 
'5.    0*  +  y  +  ,*; ;   a?^  _  2  a'  ?/  +  t/^. 


ELEMENTS  OF  ALGEBRA.  28 

6.  x*  —  2x^f/z-{-  4y-^2;  ar^ -f  3xy  +  '^^xz  +  3/  +  0  z\ 

7.  a;-2y  +  a;ijri+y'. 

8.  2  X*"  —  4  x"y"  +  2  y"^».  9.    a;"  y  —  af  "  *  i/^\ 

10.  a'^-  -  2  a^b"  +  6=^";  a^'  —  1  ~  a--^'. 

11.  3a*''  +  ^  — 4a«'' +  2a=^«-^  — a«-2. 

12.  2al  — 3a->*«  — a-i\ 

13.  Sx^  -42f-^  +  ox'-'^  —  af-\ 

14.  2  m'-^  +  3  m'-^  —  4  ?/t'-». 

15.  Saf  —  4.af-^  -\-  Saf-^  —  af-^         16.    a"'"— '  — ^("-^>"'. 

17.  2a;  +  1,  4a;''^  — 2a;+  1;    4  +  9  a^  4  —  9  w^;  4  ^  -  26, 

16a''«  +  8a6  +  4  6'^;  a  +  10,  «2-l()«  +  100;  a^  —  S, 
a*  -\-  S]  m  —  n,  m*  -\-  m^  n  +  w,^  n^  -\-  m  n^  +  n^ ; 
l--2y,  1  4.2y +  4/;  aJ-1,  a«6«+  a«6^  +  «*Z»^ 
+  ««6»+  aH^H-  a^>  +  1. 

18.  a;=^+a:-^a;2— a:-2;  WaX^'x^^—  ^^^ul'^h^x^^ -\- ^^aV'h^'^x^^ 

-  iT.?«^''*"a:'''+2^?Z>^';  a:^"  +  y-*"',  a:^"  —  a;*«y-^"' 
+  y-«-;     05'  -  y^  x^«  +  a;^^  5^  +  a:^*  y"  +  a:'y"  +  y«> ; 

3.6m  _  2.8m  y8H_,.y«« 

19.  2a«  — 3y-»,  4  .r< -f  6  a^y*  +  9y-«;  4  a*  — 3  7i-»,  16a» 

+  12  a*  n-«  +  9  w-«;  ti  «*"  ar^  +  I  ^*,  3  aA"  +  2, 
81o!-  — 54^A''  +  SGrtJ"  — 24aA"  +  16;  cx«  — ay"', 
c^a;*"  4-  ac^x^^y"'  +  a^c'^x^y^'''  +  ««^«"y»"'  +  a*y"; 
J  a^-  +  .04  6J%  I  a^"-  .04  ^J";    8"  a*"  +  9",   (64)"  a^" 

-  (72)-  a*"  +  (27)*. 

Exercise  31. 

1.  ±  5a;y^;  ~2a26a;«;  -5«i'';   ±3a*i^ 

2.  _7aW>-«;   ±|ar^y*«*;  -a^V;   i-'^^ST'- 

3.  a:*;   ±  11  a;«.y;   ±5aft;   ±2^-26^ 

4.  —  3  a"  ^.-*;  -  4  7/1  n^  a-«:  7»^  w«. 


24  ANSWERS  TO  THE 

5.  -la'-y-^',  2a3x-2;   ±^0"})^. 

6.  la^hc^'d-^',  ±^0,-^1)"',  2;  -  2  a». 

8.  £c'"^;  2a2  6*a;«;    iOir^^'^+^j  _2cc"-2/+s^ 

10.  ±  2x''y«"^'^;  — fm-^Ti-i;  a6^c-\ 

11.  \\  a  bi  c- i,  or  —  \%  a  b^  c-i. 

^   i_  13  m  —  l 

12.  V84;  5''xf;  3»aH";  a-i;  xy,  x  »    /;  icy-^;  a^b^^c^\ 

{x  +  y)  (x  -  yy. 

13.  (a  h^  c")";  «'»  {x  —  y") ;  a'^  x^p  ;  (a?  +  ^)2«. 

Exercise  32. 

1.  y  —  1;  3  a^  _  2  a  —  1.  10.  1  —  a  -^  a'^  —  a^  -\-  a\ 

2.  2a3-3a26— 5a62_^76^  '11.  Sm-zz  +  '^^^r  +  y. 

3.  x^-6x^  +  12x—S.  12.  x^-3x^y  +  3xy^-y^ 

4.  a2+2a  +  2.  13.  5a;2_  3  ^^  _^  4^2 

5.  3  +  5  X  —  2  x2  +  x8.  14.  x^-3x^  +  4:X  —  5. 

6.  ab  —  2ac  +  3bG.  15.  2  —  4  ai  +  3  M. 

7.  7  a^  _  2  a  -  |.  16.  ^x^—xy-\-^y^;  x^-3x^-2. 
3.  2x  +  3y  —  r)a.  17.  5 a:?  —  3 £c*  +  4. 

9.  m^~3am''  +  3a''m  —  a^  18.  ip^  _  2^2  _  3^-1^ 

Exercise  33. 

1.  182;  6.42;  H;  ^^;  .315;  1.082. 

2.  .5555;    75416;  30709. 

3.  .2846;     .9486;  .0316;     .3794;     .5000;    .0169;    1.8034; 

4.5728. 


ELEMENTS  OF  ALGEBRA.  25 

Exercise  34. 

1.  a^^x—1;  x^—ax—a^.        7.    a  —  b  —  2c. 

2.  2x'  +  Aax-3a\  8.   1  — x -\- x^  —  x\ 

3.  x^-2x-{-l.  9.    2x-^-3a;y +  5y^. 

4.  3a'»-2ci6~^.  10.   a  +  26-c. 

5.  l'x^-l-x-3.  11.   x"^  +  a;y-2/. 

6.  .3x^  —  xi-6.  12.   2y-  — 3x2/ +  4ar2. 

Exercise  35. 

1.  42;   32.4;  .625.  2.    ^^,  or  .0425;  .0534. 

3.  .861;  .430;  2.017;  .669;  .200;  .873;  i  ^i-i,  or  .637. 

Exercise  36. 

1.  anbi'^crl;  200 a;»  (*  —  .V")  («  +  JTY  (^  —  y  +  «")*• 

2.  —  32a\/2axy. 

3.  Sx^-2  -\-x-l 

4.  2xi''  —  4  +  3a!r-i". 

5.  4  a;"'  4-  2  aj*"  —  x'*". 

6.  a;iy-J  — 2  4-a;-'y*. 

7.  aa:»-2  6a!«  +  3c. 

8.  i  x«  -  2  X  +  i  a. 

9.  rt"  Ta;'";  a4±x*. 

10.  2^^  +  4a;y-  3a;«. 

11.  x^""  — 2a?^y"  +  4y*".         26.    3  a*  —  2  a  +  1. 

12.  X-*  — 2ar>  +  l.  27.   x-^  +  af  — aj. 

13.  2x--iy«.  28.    a- 2. 

14.  3a-J  — i  +  2a-J«.  29.    (a  +  ft)^"*  x  +  2  a"r. 

15.  1—3  a.  30.    x"  +  x"-*  +  x«-*. 

16.  X— y.  31.    2  — a»"-^ 


17. 

2a +  1. 

18. 

±12;    ±8. 

19. 

a  4-  1. 

20. 

a^-ab-{-b\ 

21. 

517. 

22. 

384. 

23. 

a^  —  3  a  +  5. 

24. 

X*  —  (m  -I-  w) 

25. 

5  a -2^ +  3 

26  ANSWERS  TO  THE 

Exercise  37. 

1.  4  c.  18.  a  +  2b  — IS  G  +  7Sd. 

2.  a  —  b  +  c.  19.  —  6  a.  20.    0. 

3.  —Sx^  -Sx.  21.  4  a2  _^  4  ^2  _^  4  c2  +  4  d\ 

4.  —11  X  -  2  y,  22.  0. 

5.  3  a  -  8  Z>  -  2  c.  23.  12  ^8. 

6.  —  35  «  +  30  ^>  —  30  c.        24.  —  3  a  7i  2/. 

7.  4  ^  -  •  16  ^  —  2  c.  25.  2  ny'^. 

8.  X  h  2  y.  26.  Saxh  —  3m  +  6  n. 

9.  3  b.  27.  0. 

10.  2xy--  'J  y  —  z.  28.  3m'^n^  ■\-2  m^  n^  —  n\ 

IX.  ^^^a  +  S.  29.  ^jy^  +  ^xy. 

12.  a  --  J-^  b  -Jr  Y-  c-  30-  S  x"^  —  8  y\ 

13.  3  a  4-  4 :?.  31.  a  (^  +  c)  +  5  c. 

14.  7x'-hy.  32.  (a  +  /*)  —  9. 

15.  210' Z>- 222  a +84.  33.  (a;  +  ?/)  +  ;^. 

16.  ^-  xh.  34.  (:*;  +  ?/)  —  z. 

^^     ^^-^CL'  35.    (a  +  ^,)2_(a+&)  +  l. 

Exercise  38. 

1     «4  _  j-^  ^3  _^  5  ^2  _  2] ;  ^5  _  [6  w2  —  3  m8  -  3]. 

2.  3:r-[22/-5^  +  4?^];  »«/>«-  [2  a^  ^^  _^  a  ^»«  -  Z.^]. 

3.  A:X-\-3ax^—\Qx^  +  Bcy~y'\\    x^  —  y^—[z^—ab  —  3acK 

(3  or  -  2  y)   +   (-  {4  7Z  -  5  ^})  ;       (a^  b^  -  2  a^  b') 
^(^_{ab^-b'}).     (4:x  +  3ax^)  -(6x^-{y-5cy}y, 

5.    -   [3ay  -  2  ab-]   -  [5  b  x  -  4.  b  z^  -   \2  c  d  -  3], 
-[3a2/-2f^^'-4Z*,t]  -  [5^>ic  +  2^cZ  +  3]. 


ELEMENTS  OF  ALGEBRA.  27 

6.  -  [-  a  +  26]  -  [rf  -  c«]  -  [1  -  ;.]  _  [aj  +  2y] 

—  [/i  —  2  //t]  —  I4:abc  —  jj],  —  [2  6  —  a  —  c  z} 

—  ld-{-l-z]  -  [a;  +  2y-2  w]-  [n  +  ^abc-pl 

7.  -  pxy-2x]  -  [5a:«/-4x^y-^]  -  [xyz-x'fj, 

—  [3  j:^  —  2  u;  —  4  x-y-]  —  [5  ic^'y-  +  xy  2  —  x*y^]. 
3.    _[_x*  +  4«»]-[3a=^-3a^]-[l-a],-[4a»-3a*-x'^] 

-Pa'^-a+lj;  -  [2  w-4j»]  -  [37i  +  l]-[5x  +  6y], 

—  [2  m-}- 371  —  4/)]  —  [5x-f  1  -h6y]. 

9.    —  [rtc  — a/i]  —  [c'x- ai]  —  [ax  +  ayj  —  [3aic  — 3x^2], 

—  [«  6*— a  ?t  —  a  6]  —  [c  x-\-a  x-\-abc\  —  [ay+  2a6c — Zxyz\. 

10.  (2a6-3«y-|-46«)-(56x-[-2c'(i-3]).     (a-2^*  +  cz) 

_|.(2_</_l)-|-(2  7;i  — X  — 2y)-(7i  — [— 4a6c-j9]). 
(2  X  —  3  X y  4-  4  X- y^)  —  (5  x^  y-  —  [x*  y'  —  xy  .^]). 
(x»-|-  3 a* -  4a«) -  (3^2 _[«_!]);    (4 j9- 2m -3/1) 

—  (5x— [— 1— 6y]).  {an-\-ab—ac)  —  {cx-\-ax-\-\ay) 
~{h^y~  [jixyz  —  'dab  c]). 

11.  m'-hem*-^?*-!- 12?^i7i-  +  8  7i»  — 3m2x  — 12mwx-12  7i'*x 

H-  3  m  x^  -I-  G  71  x^  —  x^ 


Exercise  39. 

1. 

9;  1. 

3.    4;  2. 

5. 

1. 

2. 

-A;  i- 

4.    21;  25. 
Exercise  40. 

6. 

1. 

1. 

16;  9. 

4. 

¥;  1-           7.    4;  ]^. 

10. 

3. 

2. 

12;  5. 

5. 

0.                     8.    1%. 

11. 

2;-6§. 

3. 

1;  60. 

6. 

Exercise  41. 

12. 

9 

1. 

-1.7. 

4. 

66§;  20.       6.    2. 

8. 

8;  270. 

2 

3 

5. 

5.                   7.   5;  9. 

9. 

-t;i- 

3. 

11. 

10. 

csj. 

28 


ANSWERS  TO  THE 


2. 

3. 

4. 

5. 

7. 

8. 

9. 
10. 
12. 
13. 
14. 
15. 
17. 
18. 
19. 
20. 
22. 
23. 
41. 
42. 
43. 
44. 
45. 

46. 
47. 
48. 


Exercise  42. 

12,  17.  24.  2  at  65  cts.,  22  at  35  cts. 

17,  31.  25.  25  lbs. 

$20,  $30,  $40,  $50,  $60.  26.  A,  60 ;  B,  10. 

Father,  48;  son,  12.  27.  A,  72;  B,  24. 

28.  7  years. 

29.  40  miles. 

30.  A,  28 ;  B,  14. 

31.  103  gallons. 

32.  3.7. 

33.  20,  21,  22. 

34.  24,  25. 

35.  28,  29. 

36.  100. 

37.  200. 

38.  Watch,  $117;  chain,  $68. 

39.  Linen, $0.32^;  silk, $1.95. 


1. 

A,  25;  B,  20. 

A,  65;  B,  40. 

A,  25;  B,  5. 

8,  9,  10. 

2,5. 


5. 

$50, 

Father,  36 ;  son,  12. 
100,  200. 
$23.40. 
162. 

First  kind,  21 ;  second,  42.  40.    11. 
77  at  13  cts.,  11  at  11  cts. 
21  three-cent  pieces,  18  five-cent  pieces. 
Son,  $1.04;  father,  $1.41. 
76  ten-cent  pieces,  19  twenty-five-cent  pieces. 
$70,  $52. 

25  dollar  pieces,  10  twenty-five-cent  pieces,  20  ten-cent 
pieces. 

Florins,  53;  shillings,  71. 

10  shillings,  20  half-crowns,  5  crowns. 

40  guineas,  52  half-crowns. 


ELEMENTS  OF   ALGEBRA.  29 

49.  4  children,  20  women,  60  men. 

51.  Oats,  20;  corn,  40;  rye,  120;  barley,  480. 

52.  $1225  at  7%,  $1365  at  8%. 

53.  f  180  at  15%,  $150  at  8%.  50.  30,  10,  223. 

55.  Saltpetre,  6;  sulphur,  9;  charcoal,  6.     54.  $5070. 

56.  51  women,  65  men,  150  children.  57.  $3.75. 

58.  A,  47f  miles ;  B,  37|  miles.  60.  10^  cents. 

59.  $313^  at  15%,  $16§  at  8%.  62.  7. 
61.   Horse,  $375;  carriage,  $300;  harness,  $75. 

Exercise  43. 

1.  2x2  X<iX  ax  (I  Xbxbxbxx;    2x^XxXXXxXyXy', 

[\X^XaXbxbxbxcXc\  2  X  2  X  5  X  «  X  ^  X  <;  X  r  Xc; 
5x7xa;XiFXa;X2/XyX«X«X«X2X«X«; 
^XlXaXbxbXxXX',  3x3x2x 2xaXbxbXxXxXx. 

2.  ^aHxA^a'b',  3x«y3  X  3xV;  "d  ab'^x'^  y^  X*d  ab'^x'"'  ^', 

13a4-6i  X  13ai"^>i. 

3.  4aU*;  3alMx;  8ai6ic2;  5a-ift-^x»y». 

4.  x\'X^x\x^xix^x^\  m'"  wJ^.m?",  ?Aii''.mi''?/ii''.ml''; 

a;4 .  xi .  x\  a;i  •  jc* .  a;i .  jc» ;  xl   xl-  xi,  jc*  •  a:*  •  xJ  •  xK 

Exercise  44. 

1.  n{77i  +  1);  ab(4:a  +  ftc  +  3);  3a*(a-  4). 

2.  xia-b-\-  c)',  x*y^  (39  y»  +  57  x""). 

3.  x«(5  a;  4- 7);  12bxy^  {6bx  -  7  i^  -  S  axy). 
^  2  aaf  y  z  {462  at^-^  —  5S9  s:r-^  +  616ay2). 

5.  4  a  6  (a  -  1 5  6*  +  5  c  +  2  a  6»  a;*  +  4  y  -  9  a«  c  ari) . 

6.  a;iy(2  — a6a;l  +  cajly«);  6a;J(a;  +  2a;i  — 3). 

7.  iac»(i<r-l  +  a-n-4a4&-lcl);  a"a:"(a*-- a"x''  +  x2"). 

8.  a''^"c*(ft"<r'"4- a'-ft*"  — a^^c"). 


30  ANSWERS  TO  THE 


Exercise  45. 


1.  (x  +  8)  (x  +  11);  (x  -  3)  (a;  -  4);    (a^  -  8)  {a*  -  12). 

2.  {x  +  S)(x  +  27)',  {bc-n)(bc-lS). 

3.  (aH^  +  12)  (a^  P  +  2o)  ;   («  +  11  ^)  (a  -  6  i). 

4.  (6^^-8)(a^'  +  3);   (^^  +  4)  (t/^HH);   (aH5)(a«+12). 

5.  (ab-]-2c){ab—5c)',  (a'' +10)  (a^— 12);  (n  + .5)  (71  + .3). 

6.  (^'^+762>)(a2-8^^^);   (x+5)(x-U);  (x''-\-25a'"){x''-12a'). 

7.  (i:c-4)(aj-ll);  (m  +  i)(m  +  |);   (oj  +  2)(x- 13). 

8.  (a&  +  5)(a&  +  26);    (a-5^>:z;)  (a- 15^»ir) ;    (^/HSa;^) 

{i/-9x^)',   {l  +  i)x)  (l  +  7x);   (m-7a){7ri-Sa). 

9.  (a  +  9x1/)  (a  -  21  xy)  ;  {x  +  ij  +  4.)  (x  +  y  +  1), 

10.  (1 -5a&)(l-8a6);  (a  -  ^  +  2)  (t^  -  ^  -  1) 

11.  (x-y+2){x-y-b);   {x  +  21)  {x  +  21)',   {x'-12)  {x^-ll). 

12.  [(a  +  Z^)2  +  1]  [(a  +  ^)2  +  8J  ;    {x^^'-b){x''--m). 

13.  (a  +  3  ^>2  c)  (a  -  13  b^  c)  ;  (ic«  +  a)  (x«  -  b). 

14.  (ic  +  5  2/)  (a;  -  14  2/);     {X  +  1)  (a;  -  I);    (x^"  -  20) 

(cc2«-23). 

15.  (aj  +  1)  (oj  -  1)  ;   (x'  +  21  C.2)  (^2  _  22  ^2), 

16.  (x  7/  +  11)  (xy  —  14)  ;   (a"  a;^'"  +  11  if)  (»"  :z;-'"  -f-  3  ?/«). 

17.  (a;  ?/  —  11  a'^  6'*)  (a;  y  —  17 a''^*")  ;   (a;  -  ^)  (cc  -  1). 

18.  (a;2»2/'"+17a'"6'")(£c2«^2«_^3^m^.„-).       |-(j^_^^)3m_|_7^4nj 

[(aJ +  ?/)'"•- 14  a^"];   {n'+  .11)  (^i^-.l). 

19.  (o^  +  f )  (a^  -  I)  ;    (:r  +  2.1)  (:r  -  .1);   (a'  +  1)  (a^  +  i). 

21.  (2  cc  -  2)  (2  X  -  3) ;     (3  a:  -  3)  (3  a:  -  6) ;    (2  :z;  +  6  a) 

{2x  +  2  a). 

22.  (Sa  +  4.b){3a  +  6b);   {4.x-2a)  (ix-  3  a). 

23.  (5  x^""  +  J  a")  (5  x'^'"  -  i  «")  ;  [6  (a  -  b)^"  +  13  (a  —  b)'] 

[6{a-b)^"  -  11  (a-b)^]. 


ELEMENTS  OF  ALGEBRA.  31 

Exercise  46. 

1.  (4x+l)(x  +  3);  (2y+l)(2y-3);  {3a^-\-x')  (ia^-x^. 

2.  (3-a:)(H-4-c);    (4  a;  +  3y)  (2a;  -  7  y);    (3aa:-l) 

(2aa:-f  1). 

3.  (?/i'-3)(8w»  +  9);   (3a- ll)(r>a- 1);   (2a  +  3Z>) 

(3  a  —  6) ;  (2  m  —  n)  (m  -  G  n) ;  (3  a;  +  4)  (a:  +  1). 

4.  (8  +  9a)(3-8a);    (a:+ 15)  (15a:-l)  ;  (44-3a;)(l-2a;). 

5.  (3x~2y)(2a:-5//);  (4x-3y)(2x+5y);  (a:-5)(15x-2); 

(12  x  -  7)  (2  a;  +  3) ;  (a  +  3)  (11  a  +  1). 

6.  (3-5x)(6~x);  (3x  +  y)C2x-3y)',  (l  +  7x)(5~3x). 

7.  (8x  +  i^)  (3x  -  4y) ;  (2  x2«  +  7  y*")  (3  x'^"  -  y^). 

8.  (x4-y  +  2w  +  2/i)(2x  +  2y  +  7?i  +  7i);  (x  +  4)(2x— 7). 

9.  {x  +  y^3a-3b)(2x  +  2y-a-b)',  (x+|)(Ya:-l). 

10.  [(x-y)»--2x"'yi«][ll(x-y)»''-x-y5"];  (3a+l)(9a-l). 

11.  [2  a"  +  3  (x  —  y)'""]  [2  a"  +  7  (x  —  y)""*]. 

Exercise  47. 

1.  ^m2H-7i2)2;   (;,i_|.,i)2.    (^  a'' -  W  b  cY',  a^tt  -  2)^ 

2.  (7  w»  -  10  7i'^)2;   (9  x^  y  -  7  a')^ 

3.  (7ii»  -  7*)^  (1  -  T)  wi7i)»;  x^  (x  +  1)'*. 

4.  («  +  />  +  8)2;   (7/1  +  0)^ 

5.  x*(2«»-5xy)2;  (19  a^»c- 2  rf  77i7z)«;   (11 7/i7i2_  IOje?)^ 

6.  (15x2~y2)2;    {2a"'-b'''y. 

7.  n«(7  7w  +  3^)^   (:^+  i)^ 
8-  (s«*+  i*')';  c(aM-3fc»)«. 
9.  {3x-},yy',   {m--7i-\-iy. 

10.  (a2-a  +  3)*;   (2x  +  2y-f  1)^. 

11.  (ai  -  il)«;    (;/ii  -  1)^;  mn  (ml  -  niy. 

12.  (xJ  +  yi)«;   {m  ni  -  a)«;   (2  xJ  +  3  /t)'. 

13.  i(a^br-r,rY-  {^,xl--^)\ 


32  ANSWERS  TO   THE 

Exercise  48. 

1.  (l-7x)(l  +  7x  +  ^Qx')',  {2x-9f)(4.x''+36xf  +  Sly'); 

(6  x''-a)(86x'  +  6ax  +  a"). 

2.  {xy  —  ah)  {x^y^ ^r  cuhx^y"^ ^-  a^h^x^y" -\-  ahx^if  \x''y')', 

\x  _  1)  (a;6  +  a;5  +  cc*  +  a!^  +  a;2  +  ^  +  1) ; 
(3  a  -  Z>)(81  a*  +  21  aH  +  9  a?h''  +  3  a  ^>«  +  h')-, 
(a  b""  —  m)  {a"  b^  +  aPm-{-  in''). 

3.  (6a-7)(36a2  +  42a  +  49);  8x{l-3x){l  +  8x  +  9x''). 

4.  (a8  -  4  ^2)  (^12  +  4  aH^  _^  16  a«  ^>*  +  64  a^^*^  +  256  h^) ; 

(9  x-12y)  (81  c«2  +  108  ic^  +  144  ^2)  ;  (^^-i  _  y-i) 
(a;-*  +  x-^y-^  +  ^-^^/-^  +  x-^y-""  +  y"'). 

5.  5ir2(3a;-4)(9a;2  + 12a;  +  16); 

2  a6  (a  -  2)  (a^  +  2  «»  +  4  a2  +  8  a  +  16); 
(ar-i  —  2/~^)  (ic-?  +  x-^  y-^  +  2/~^). 

6.  (ab-xy){a^b^-{-etG.);   (4.a^-5b)  (16 a* -{-20 aH  + 25 P); 

(iC"  —  2/"^)    (^2«  _j_  j;c«ym  _j_  y2my 

* 

Exercise  49. 

1.  (2  a  4-  1)  (32  a^  -  16  a»  +  8  a^  -  4  a  +  1) ; 

(l  +  ic)(l-a;  +  a;2-ccHa;^);  (ic'+2/')  (^'-ic^2/'+2/'); 
(^'  +  2/')  («'  -  :e'z/'  +  ^^^  2/'  —  x^  y^  +  y^). 

2.  (a  4-  2)  (««  _  2  a^  +  4  a^  -  8  «»  +  16  a^  -  32  a  +  64)  ; 

(a;2+  Oy)  (cc*— 9x2^^+81  ?/') ;   (4.t2+2/2)  (16a;^-4a;2^2_j_^^). 

3.  (ab  -\-  x^y^)  (a^b'  —  a^b^xy  -{-  aH'^x'^y'^  —  abx'^y^ -\-x^y^); 

(x^  +  4  y^)  {x'  -  4  x^y^  +  16  2/4^). 

(10  cc  +  11 2/)  (100  x^-110xy-{-  121 1/'). 

4.  (x^  +  y^)  (£ci2  -  cc^ 2/'  +  2/''),  ^i'  (^'  +  2/')  (^'  -  a;'  ?/"  +  ?/') 

(a;i2  _  x^  if  +  2/1^);  5  x!"  (3  ic  +  4)  (9  tc^  _  12  a:  +  16)  ; 

(a;8   _|_   ^8)  (^.^16  _  ^8  ^^^8   _^    ^16^^ 


ELEMENTS  OF  ALGEBRA.  33 

5.  {X-'  -f  y-')  (X-*  -  x-^y-'  +  x-2y-2  _  etc.); 

{x^-^y'-'Xx'^^-xh/^-y^) ;  (xy-^ab){x*y'-abxY-\-etc.) ; 
(a'  -I-  n  (a"  -  a'  i"  +  6»«). 

6.  (a"  4-  *")  (a"  -  a"  4"  +  i**) ;     (1  +  x')  (1  -  X*  4-  «») ; 

(af>  +  y2«)  (x2»  -  x»y2"»  +  2^"*);  . 
(ar-»  +  y-*)  (x-i  -  x-iy-^  +  yl). 

7.  (a*"   +    6»"')  (a^"   -   a*"  6»'»    +    ^"'*);      (2a^»c  +  3  a;) 

(16a*b*c*—2-iaH'^c''x  +  S6a'b''c^x^—54:abcx^-\-Slx*)', 
(4a  +  i-^)  (2o6a*  -  64a»*^  +  16aH*-4:aH^  +  6«). 

8.  (4x'^4-9a«)(16a;*-36a«a;2  +  81a*); 

a  «'  +  J  *')  (^  «*  -  ^'ff  «'^'  +  tV  ^*); 
(a«  +  6  c)  (a*  -  4  aHc  +  11  ^^^c^). 

Exercise  50. 

1.  (ax  +   by)  {ax  -   5y);     (4x   +   3y)(4:X   -  3y); 

(5aa;  +  7  by^)  {5ax-7by^). 

2.  (ar^  +  5^)(x+y)(a:-y);  («^  +  9y2)(x  +  3y)(aj  -  3y); 

(x*  +  y*)(ar^  4-  y^)(x  +  y)(x-  y)-,    {x'  +  y)(a;*  +  f) 
(x'^y){x^-y). 

3.  (aH^  +  9a;V)(«'*  +  3a:yi)  {aib-Zxyi)-, 

(l-\-10a*b'c){l-10aH^c)',    (4^8+ 3  6«)  (4 a* -3^"). 

4.  (3a''  +  2ar»-)(3a"~2x^-);    (^  a  +  ^  6)  (^  a  -  ^  6)  ; 

(a;J  +  y*)(a:i-yi). 

5.  (a;-«+y')(a;-'4-y)(x-*-y);  (a4-^+c4-<^)(a+^-c-rf); 

(x— y  +  a)  (X  — y  —  a)- 

6.  (a4-a:—y)(a—a:+y);  {ab-\-xy-{-l)(ab-\-xy—l)',  4ab. 

7.  (a  4-  Z^  +  2)  (a  -  ^») ;   (a  4.  6)  (a  -  *  4-  2)  ;  503000. 
a   47a;(aj4-2y);  2805000. 

9.  12  (a;  -  1)  (2  a:  4-  1) ;  1908  X  1370. 

3 


34  ANSWERS  TO  THE 

10.  (a^«  +  1)  (a^n  +  1)  (a«  +  1)  (a»  -  1) ;    xy{Sx  +  y) 

(9^2  —  Sxy  +  f)  {Sx  —  y)  {^  x'  +  3£cy  +  t/^)  ; 
h  {a"  +  h^)  (a  +  b)(a-b)',   {ah  +  bh)  (ah  -  bh). 

11.  6t  (a  +  4  a;  —  6);       (2  aj-^  +  3  2/"^)  (2  a;"^  —  3  5/-1) 

(4 a;--  —  6  a;-*j^-i  +  9 y-^)  (4 x-'^  +  6  a-^y-^  +  9  y-^). 

12.  150000;  2{ah-\-2hx){ah-2hx)\  (5a4"+3i^>'«)(5a5"-3i^>'»); 

(a:  +  y)  (a;  -  2/)  (x^  —  a; y  +  y^)  {x'^  j^xy  +  y^. 

Exercise  51. 

1.  (a  -^  b  +  c)  (a  —  b  —  c)-^  (a -\- b  —  y)  {a —  b  —  y)-, 

(^a-b  +  2)  {b-a  +  2). 

2.  {5x-\-b-{-3c){5x—b  —  'dc)\  (a-\-x  +  y-\-z){a-\-x—y—z). 

3.  (2a;-3y+9)(2a;-3y-9);  (a;  +  3)  (a;  +  4)(a;2-7a;-12); 

(2  a:  -  1)  (a:  -  1)  (2  a;2  +  3  a:  -  1). 

4.  (4a;2+a;-^)(4a52-a;+i);  (3a+:r+42/-l)  (3«-a;-42/-l). 

5.  (a;  +  5/  —  m  +  w)  (a;  —  y  —  m  —  7i) ;    (a^  +  ^^  +  c^  4-  ^^) 

(^2  _^   ^2  _  ^2  _  ^2^  .    4  ^2«   (^n  ^  J«>)    (^«  _  J«)^ 

6.  (;s  +  2a!— 32/)(^— 2a;  +  3y);    (2a  +  l— 2a;)  (2a-l  +  2a;). 

7.  (x  +  3^  —  ^)  (a;  —  2/  +  «)  (ic  +  .y  +  ^)  (a;  —  y  —  «) ; 

8.  (c  +  (^-3a  +  2a;)(2a;-3«— c— c^);     {2x—Sy-{-4.z-\-bd) 

{2x-Sy-4.z^5d);    {b  +  c  +  2x)  (2x-b -c). 

9.  (5a«+4a2+a;2-3)(a;2+4a2_5a84-3);    (y-^bb  +  Sbx-^l) 

(3,_5J_3^a;+l) ;  a2„^^«.|_2)(a"-2)  (a2«_6)(a2'  +2). 

10.  (3  a  4-  *  +  ^"^  —  y"*)  (a;"  —  y*"  —  3  a  —  J). 

11.  («»  +  a^^"  +  y  -  3  ;s)  (a^  +  xS"  +  3  ;s;  —  2  ?/2m). 

12.  (2aj  +  3?/  — 67?.  —  4^)  (2a;  — 3y  +  6  71  — 4;?). 

13.  (a''  —  6«  +  c"*  +  ^^m)  (^«  _  j«  _  c"*  —  A;2'»). 

14.  (2a  +  3a;  +  4?/  — 8«)  (2a  + 3a;  — 4?/  + 8«). 

15.  (a  +  lb-3c)(a  +  ^b  +  3c);  (a^+  a-b"-- 3)  {a''--a-b''-\-3). 


ELEMENTS  OF  ALGEBRA.  36 

Exercise  52. 

1.  (3a^  -^  3  ab  +  2b^)  (3a^  -  Sab  +  2b^',  (a^  +  3a-{-9) 

(a'-Sa  +  d);  (-ix^ +  2 xy  +  y^){4x*-2xy  +  y^). 

2.  (x'  +  xy  +  y^(x^-xy  +  y^)(x*-x^y^-\-y*); 

(9  «'  +  10  a  a:  +  4  x^)  (9  a^  -  10  a'x^  +  4  x^) ; 
(m*  -\-  m7i  -{■  n^)  (m*  —  mn  +  n^). 

3.  (2  X*  4-  2xy  +  Sy")  (2  x^  ~  2  xy  +  Sy^) ;  (a*  +  aH  +  b') 

(a*  -  a^6  +  6*);  (9  a"  ^  6  a  +  4)  (9  a«  -  6  a  +  4). 

4.  (5a»  +  7aUi  +  46»)(5a»-7ai6a4-4^»«);  (aj  +  ^iyi+y) 

(x-iciyi  +  y) ;  («»  +  a;3  yJ  +  y^)  (a:»  -  xi  yi  +  y»). 

5.  (4a*+4a^^>i  +  36»)(4a*-4a'*^>i  +  3i»);  (3tt^+2a62+76*) 

(3a«  -  2a62  +  7^*);    (^  +  jo*  +  1)  (/>  -  p^  +  1) 
(;?«+;?  4-  1)  ip"  -P  +  1)  (i?*  -i?*  +  1). 

6.  (7aH4a6  +  96^)(7a'^-4a*  +  96*);    (3a;H3a;yH5y*) 

(3x^-3xy^  +  5y0. 

7.  (m**  +  m-  +  1)  (ttj'"*  -  m"  +  1)  ;    (x^^  +  4x»  +  16) 

(x«"-4x"+  16). 

8.  (a  4-  a*  **  —  ft)  (a  —  a^b\  —  b)\ 

(a«"  +  2  a-6'"  -  ft^"*)  (a*"  —  2  a" 6™  —  i^'"); 
(5  m*  +  2  m  »  —  4  71^)  (5  m'*  -  2  m  n  -  4  7i») . 

Exercise  53. 
1.    (a  +  ft)  (a  +  c) ;   (a  c  +  rf)  (a  c  —  2  ft). 
a.    (a-.ft)(m-«);  (a-ft)(4x-y);  a(a  +  l)(a=' +  1). 

3.  (2x~y)(3a-ft);   (;,  +  ?)  (r  -  3). 

4.  (x  —  y)  (a  —  2  ft  —  4  c). 

5.  (a  -  ft)  (5  a  4-  5  ft  -  2) ;  (2  X  +  y)  (3  X  -  a). 

6.  (2x-l)(x«4-2);  (ax  -  1)  (a»x»-ax -1);  (a;-2y) 

{m  —  n) ;  (a  4-  x)  (4  x  —  a). 

7.  (x  +  my)  (x  —  4y) ;   (a  —  x)  (4  a  —  4  x  +  5). 


36  ANSWERS  TO   THE 

8.  (a  —  c)  (S  a  —  b)  ;   {a  -\-  b)  (ax  +  b ^  -{-  c). 

9.  (ox  +  Sy)  (ax  —  bt/);  (m  ~  n)  (n  ~ p). 

10.  (m  —  n)(m  +  n  -p);  (2^/  —  3£c)  (3?/  +  x)  (3  i/  -  a:). 

11.  (c  +  7)  (3  a  -  7  6  -  5)  ;  (ic  -  2  2^)  (:r  -  3  2/  +  3) ; 

(a,  _  1)  (;^2  _^  i^>^ 

Exercise  54. 

1.  (a-\-b  +  cy.  5.    (a-b  +  c-  df. 

2.  (a-5-c)2.  6.    (3  0^2/ -4  ay. 

3.  («^  +  6  -  c)^  (oj  +  2)^;   (^  a  -  3  6  -  t)^ 

4.  (a  —  3  a:)^  7.    (m  —  ?i  —  ^  +  a;)^. 

Exercise  55. 

1.  10  ((K"  +  1)  (af*  _  4);   (^2  _^  £c  +  1)  (a;'  -  a;  +  1); 

12(xy+l)(xy-^). 

2.  (a:  -  .5)  (x  -  .06);  (.^  +  f)  («  +  ^);   (3  -  a:)  (2  +  x). 

3.  3m27i(m  +  ^)(^  -  m);    2  (2  a  -  1)  (4.0^  +  2a  +1); 

(a^  +  9)  (a  +  3)  (a  -  3)  ;  6a;«(£c  +  6)  (a:  +  2). 

4.  (xy  +  t'^)  (^2/  -  t);  C^^'^*  +  <^o)  («'^  -  f); 

\.(a  +  by^i]l(a^.by-^l 

5.  (a-x)(«'.-a:-4);  2(x''  +  x^2){l-x)(2  +  xy  a(ha'+l). 

6.  (a^3+  f )  (a^«-^) ;   (a-+o,)  (^^-+1) ;  (| a'  --3  a«)  (|  a^  m^g  a^). 

7.  (m-a)(w-7i);   (a4.8)(a-l);    (a-b)(4.a- U- 2)-, 

(la""  b^^  +  a;^)  (7 aH^""  +  3yi). 

8.  (17  +  a)  (12  -  a) ;   (x'-  -  if)  (a;^"  -  i ). 

9.  (tw  +  w)  (m  — ?i— ^);     «»  (a;  -  1)  (a:^  +  1)  ; 

a'  (l  +  b)(l-b)(l-b  +  b^)  (l  +  b  +  b^y 

10.  (20+a;)  (19~a;) ;  (a^-l)  (8a^-l)  (a^"'+  ^3'"+  ^2'"+  a-+l) ; 

l(x  —  yy^  +  5  ^2-]  [(a;  -  2/)^''  —  5  Z'^'"]. 

11.  (3a;-ll)(2a:  +  7);  12(a:+7)  (aj  +  2);  (x  +  y)  (x  +  7j-5); 

(f  a:  —  J^??^7^)  (fx-  ^  ?/). 


FXEMENTS  OP  ALGEBRA.  37 

12    (H-^3^ar)(l-rVa:);  (2ar-2/)(ar  +  3.v-2a). 

13.  (a"  +  b'x)  (a^  +  />";/);  (a-  -  (i)  (a;  +  2«  +  i)  ; 

(9x-^  +  liajy  -  y-)  {'^x'  -2xy-  y«). 

14.  b  (a*  +  ^^)  (</«  -aH^  4-  0')  ;  (a;  +  y)»;  (x^"  +  aZ»  +  ac) 

(ar-^"  —  w  ^  +  ^  c) ;   (ni  -{-  n)  (m*  —  ?/i  w  +  /i'-  +  1). 

15.  (rt  +  ^— c— r/)(a-/>  +  c— </);    (a— y  +  a;  +  2)  (a— a  +  y+2). 

16.  {x  -^  0)  {x-\-  a-i-  b)]   {x^"  +  2  a)  {x^"  —  a  —  b). 

17.  2  (57/i  +  ii^  ±  1)  (25r?iH  50//in  +  25w^  if  5m  qp  5w  +  l); 

(4w  +  70(4w-+2m7i+77*2'),  3w(12w^+8mw  +  37r). 

18.  (^c*af  —  a*)  (/•'•(j^x"  —  1);     {7  p''  +  13;;^  +  11  q^) 

{Ip'-  V6p  n  + 11  q^) ;  2  m  {m'-\-  3  7^,  2  w  (3  t/i'H  n^). 

19.  (m— 7i)  (2m— 271  +  1)  (27?i— 271— 1);    (m  +  7i)  (2??i  +  7i), 

71  (m  -f-  n). 

20.  (a-  +  a  c  +  />  c)  («  —  a  6  —  «  c) ;    (8  m*  —  4  m  ri  +  9  n^) 

(8mH4m7i+97i2) ;  (r)a;2+3xy+4y^)  (5a:2-3a;y+4y2). 

21.  x(3a;  +  2i/)(2a;  +  3y);  ^/'(Sar  +  4y)  (2a;  -  3y). 

22.  (a:»  — 3ic)(a:"  — a^»  — ac). 

23.  {m^^-\n^){m±2n)\  (4m  +  37i  -  3j9)  (3w  +  3/>  — 2m). 

24.  (x»«"  -  c)  (x""  +  a  +  i); '  (X  +  y  +  2)(a:  +  y  -  3); 

(6ar»  +  4  a;  y  4-  4  y*)  (5x^-4  a;  y  +  4^). 

25.  3xV(3«  + 2y)  (a;  — y);   (m  -  37?.)(m  +  2  ti  ±  4). 

26.  m\n\{ab—xy--^z)(,iv'b''-^abxy-\-?^xyz-\-?>z^-\-x'%f)\ 

(9  a-  -f  aJ"  b^"^  —  11  i'")  (9  a"  —  «»"  ii*"  —  11  A"*)  ; 
(9  a*»  +  3  a-i^™  —  5  ^»*'")  (9  a«"  —  3  a»  6^*"  —  5  i*'"). 

27.  2(3x-2y)(3x-2y±6);    2  (m  +  37»)  (m-27t- 6a); 

a«(a''x^+4n^)  («*x*-4a«7i2x-^+16a*7i^x*),  a^(ax-}-2n) 
{ax—2n){a^x^—2anx-\-4?i^)(a^x^-\-2ftnx-\-4n''). 

28-    (x  +  7y)(aj-4y  +  4);  2(1  -  3  a  -  2^)  (// -  x). 

29.    7»77  (m  +  7i)  (m  —  ti)*;  —  a  (a  +  m)  (a*  -\-  2am  -\-  2m*). 


38  ANSWERS  TO  THE 

30.  (3x  —  4:y)(5x-\-4:y  —  5a);  (a  —  b)  (a  —  c)  {c  ^  b). 

31.  (a  -h  c)(c  —  a)(cd  —  l)(c^d^  +  cd  -{-  1);  (mn  ±  8) 

(w^Ti^  qp  8mn  +  64);  6  w^  (4m  +  3  w)  (w  —  2  w); 
(a  a;  —  3  Z>  y)  (a  —  i/) . 

32.  (m  -  2  7i)  (m  -  3  w  +  16)  j  a;^  (3  a;  --  1)  (a;  -  1). 

33.  (m  —  n)  {6  m^  +  5  mn  +  7  n^). 

34.  9?«.'(a2+ w^),  9m'(a  +  m)(w— a);  (x+42/)(x  — 42/  +  1), 

(x  -  4  y)  (a:  +  4  2/  +  1 ) ;  (a:  -  2  a:  2/  +  2)  (a:  -  2  xy  -  3) . 
Ten. 

35.  (36a:-132/)(18ar  +  29i/);  m(m+ 1)  (m-1)  (m^-w^-lO). 

36.  vi{m-\-n){m^  ■\-mn-\- n^){m^—mn-\-'nP))  {y-\-\){x—l) 

ix-y^-l);  {x-\-2)Hx-2Y. 

37.  (?/i  +  4)(m4-5)(m  — l)(m  — 2);  (3  —  w)(ww  — w  — 3). 

38.  (x»  —  ^  —  a:-")2;   {x-'^ -^  y-^)  {x-^  —  y-\). 

39.  7o2a:(a;-2a)(2a:-a);  (x-«+y-«)  (a:-*  +  2r*)  (a^*-2r*). 

40.  a:2(12a:8-8a'2/H2l2/);  a:i(4arJ±3)  (16a:q:12a:i+ 9). 

41.  (a:  -  y)  (a;  —  2  ?/)  {x  +  ari^/i  +  2/)(a:  —  ariyi  +  y)  ; 

(a;-  +  1)  (x"*  +  2)  (a:-  -f  4)  (oT  +  5). 

42.  (2a  +  3^')  (2a-3/>')(ar-2a)(a:2  +  2ax-f  40^^); 

{m  +  2n-^p)(m  +  2n  —p)  {p  +  m  —  n)  {p—Di  +  2w) ; 
(a:-  +  i)  (x-  -  i)  (a:^-  +  ^V)  {^""  +  tV);  («'"  +  *'") 
(a*"  +  5»)  (a*"  -  b^)  (a*"'  -  b*''  -  6  a^"*  ^»2«), 

43.  (x^  +  2/')  (a:  --  2  I/)  (ar^  +  2a^y  4-  4  y^)  {x^  -  x^y''^y')\ 

(a;"*  +  1)  (ar^"*  +  4)  (a:^"'  _  ar«  +  1)  (a:*"'  —  4  ar^*"  -f  16). 

44.  (^-\-n— p){m-\-p  — n){m-\-n-\-p)  (m  —  n  —  p). 

45.  4  (a:"*  +  2)  (2/"  +  4)  (a:-  -  2)  (i/"  -  4). 

46.  (a:"*  +  1)  (a:-"  +  2)  (x^"*  +  4)  {x^  -  2)  (a;^"'  _  a:*"  +  1)  ; 

(2  a:"*  +  3)  (4  x""^  +  9)  (2 a:"*  -3)  (a;"»-  1)  (a;2'"  +  a:"»  +  l). 


ELEMENTS  OF  ALGEBRA.  39 

Exercise  56. 

4.  6a«*x*;  2axy.  13.   x  {x  -  y).  21.  4rz-l. 

5.  Za'^^y^  !*•    ^-^^y  +  y'''  22.  x  +  2. 

6.  6x^2/-^z^  15-    4(a-^).  23.  x  -  3. 

7.  2a:iyJ.  16-   4;s(a:-y).  24.  m  -  n. 

8.  6(«  +  ^)-  ^^-    2x-3.  25.  or -2. 

9.  ^y^^.  18.    x^'Sy.  26.  x^^-e. 

10.  a:*(3x+2).  19.    x  -  y.  27.   a:"  -  5. 

11.  3a«x-12a».        20.    x^  -  x.  28.    2a:"-5. 

12.  x  +  l;    a:" +  6;    x  +  3. 

Exercise  57. 

1.  ar«  _  3  X  4-  2.  9.  ^^  (^  —  3). 

2.  a:^  — 2x  +  l.  10.  9m»(wi-l). 

3.  ar»  +  2  X  +  t.  11.  a:  -  y. 

4.  (x  -  1)  (x  -  3).  12.  mn{x^-  3). 

5.  x{x-  a).  13.  x^  -  X  —  1. 

6.  X  —  y.  14.  2  X  —  5  y. 

7.  X*  +  2  X  +  3.  15.   2  n  (m^  +  4  m  y  +  7  y'^). 

8.  3  X  +  2.  16.    2  m"  x"  (x«  -  1). 

Exercise  58. 

1.  2(x  +  y).  3.    X4-2.  5.    x"  -  2.        7.   x-2y 

2.  ar»  -f  X  +  1.        4.    2  X  +  3.        6.    2  x^  +  5. 


40  ANSWERS  TO  THE 


Exercise 

59. 

1. 

x'-Vy. 

17. 

x''  -  1. 

2. 

x-y. 

18. 

71  (n  +  x)  (w  —  x). 

3. 

x-1. 

19. 

x  —  2m. 

4. 

x-1. 

20. 

xyi^-yY' 

5. 

x-'d. 

21. 

a-b. 

6. 

'6x'-  +  1. 

22. 

a  -{■  b  +  c. 

7. 

x^  +  ^y  +  /. 

23. 

X"  +  2, 

8. 

x+l. 

24. 

x-2. 

9. 

x{x-^  b). 

25. 

2{x  +  y). 

10. 

2x'--^. 

26. 

7^2_^8x.+  l. 

11. 

a"  -  h\ 

27. 

n  +  2. 

12. 

a-h. 

28. 

3m(y'+4f-2y  +  S). 

13. 

3a:2«  +  2m2. 

29. 

x''-Sx-\-  1. 

14. 

(m  -n)(x-  y). 

30. 

2(:r+l). 

15. 

x^  +  A. 

31. 

x--2y\ 

16. 

a"  +  b"^. 

Exercise  60. 

1.  S19axU/z^  10.  (n-xy(n*  +  a^x''  +  x^). 

2.  lUm^n^x'z^  11.  x^-6x^  —  19x  +  S4:. 

3.  aea^^-^cl  12.  105xy^x^-y^). 

4.  72  m^n^y^.  13.  :r^  —  1. 

5.  12aic3y4(a?2-3/2)2.  14.  (3a:+ 2)(a^  +  2)  (;r+ 3). 

6.  m'^n^  (x^  —  y^).  15.  {a  ^  x){b  -\-  x)  {c  +  x). 

7.  12  a:iy2  (a,2  _  y2>)  j^g^  3mn  {x  -  yf  {m  -  n). 

8.  (x2-16)(:z;2_25)(^_6),  17.  („4.^)2(^2_^2). 

9.  x(x  +  2y{x+l){x-\-'S).  18.  ^i^-l. 
19.  (a:^-/)(x2_^2). 


ELEMENTS  OF  ALGEBRA.  41 

20.  3a«x(3a:-a)  (2a:  +  3a)  (a:  +  5a). 

21.  (x  +  4)  (X  +  3)  (X  +  1)  (x  -  2). 

22.  (x-y)(3x-2y)(4x-6y). 

23.  a*  -  1()  6*. 

24.  {a  -I-  ^)  (w  +  w)  (x  +  y). 

25.  (4  a;  -  5y)  (2  JB  -  7  y)  (x  +  y). 

26.  (x^  +  y^)  (x*  -  x^y*^  + /). 

27.  20x2y(3x+l)(5x  +  l)(4x-l). 

28.  (a-f ^  +  c  +  d)(a  +  ^'  — c  — c?)(a4-c  — i— e/)(a  +  rf— Z»  — c). 

29.  x*  +  xV  +  y*- 

30.  6x2(x  +  7)(3x-f  5)(3x-2). 

31.  12x-(x"  +  2)(2x«  + l)(4x"-7). 

32.  abc(m  —  n). 

33.  (a  —  b){b  —  c).  34.    ale  {a  —  x)  (^»  —  x)  (c  —  x). 

Exercise  61. 

1.  2x*4-x«-17x2-4x-f  6. 

2  and  3.     x*  +  5  x*  +  5  x'^  —  5  x  —  6. 

4.  (x-2m)(x  + m)(x*  +  m*)(3x2-mx  + m«). 

5.  2xy*(30x»  +  95  x*y  +  68x»y»  +  32x«y«  +  24xy*  -  15y»). 

6.  X*  -  14  x«  +  71  x**  -  154  X  +  120. 

7.  (x2-3x  +  2)(x*  +  3x*-8x«  +  40x-96). 

8.  ar*  +  2x»-9x^-2x  +  8. 

9.  3(6x*  +  ar*~33x»  +  43x«-29x+ 12). 

10.  6  X  (x  -  1)*    (.r  +  1)«. 

11.  x*4-5x»4-5x*  —  5x  —  6. 

12.  2x>-2x'-3x«  +  3x*-2x«-3x''  +  2x  +  3. 


42  ANSWERS  TO  THE 

Exercise  62. 

1.  (a  -  ft)  (a  4-  by  (a^  -  4:b^)  (a^  -  ab  +  b^). 

2.  x^"  +  7  x^«  —  10  x^''  —  70  a;2"  +  9 a"  +  63. 

3.  (a;"  4-  4  2/"')  (x"  —  2y^)  (x^"  —  2a;"2/;«  +  S?/^'"). 

4.  2(x  +  3)  (2  X  +  3)  (x^  -  1)  (x^  ^x^  +  l)(x  +  2). 

5.  xy{a^x)(b-y){2b-y)(2b-'-xi/). 

6.  a*"*  —  b^"".  7.    x*"  —  16  a**". 

8.  (ic"  +  c)  (2  x"  —  3  ^»)  (a;2»  -j-  a  a;"  —  ^»2). 

9.  (ic  +  2y  (x'  +  4)  (X  -  2)  (x  ~  3)  (x^  -  16). 

10.  (X2«  ~  a^)  (aj2«  _  ^2)  (^2«  _  ^2^  (^6«  _  ^6-^^ 

11.  Sx  —  y,    (3x  —  y)  {x  +  yf  {x  —  yy. 

12.  3^2  -  2  <   3  ic^  y  (3a;2  -  2  a^)  (a;  -  9  a)  (2  x  +  5  y). 

13.  2  a;"  +  1,    (2  x«  +  1)  (x^"  -  1)  (9  x""^  -  4). 

14.  The  expressions  are  prime  to  each  other,  (a*  -\-  a^b^  -f-  b*) 

(a  +  by  (a  -  by. 

15.  a;  -  5  ^>,   6  (a;  -  5  ft)  (x^  -  9  ««). 

16.  x^'*  —  7  a;"  +  12,    (a;2«  -  7  x"  +  10)  (a;^"  —  7  a;"  +  12). 

17.  c  a;  +  *^    (c  a;  +  ^^)  (a^^  -  c^)- 

18.  m^  +  a;  1/,    (w'^  -\-  x  y)  (4:  x^  —  9  y^). 

19.  a^m  ^  ^2n^      (^2m  _|.  ^2  «^)   (^2«  _  4  ^2m)^ 

20.  a;^  +  X  1/  +  y^,    (x*  -{-  x^  y^  +  y*)  (x^  —  4  y^). 

21.  5a;2  -  1,    (5a:2  -  l)2(4a;2  +  1)  (5x'  +  x  +  1), 

22.  a;"  —  y"*,    (a;«  —  .?/"•)  (a^^"  +  x^^i/^'"  +  y*-^), 

23.  a;''  -  8  a;*  +  50  a;2  -  a;  -  42. 

24.  (x''  +  7x-^  12)  (a;2  +  0!  +  3)  (a;  -  2)^. 

25.  (2  a;*  +  5  ^2  ^  3)  (4  x*  -  49). 

26.  x(6x^-  31  x*  —  4x^  +  Ux^  +  7x-  10). 

27.  a^s  +  3  a;^  -  23  .t^^  -  27  x^  +  166  a;  -  120. 

28.  3  a;^  4-  2  a;^  -  3  a!  -  2. 


ELEMENTS  OF  ALGEBRA.  43 

Exercise  63. 
2  2x  —  3y 


2bxy'    37/i«aj«/'         2x      ' 

«     .  n(x*  —  y^)      2  X  +  1 

m  X  —  J 

^    3m  +  2     ..        ^    y"-»       „    *  +  a?     3  +  a 

^-  3^;^^  ^(^-y>^  ^-  °-  r-f^'  -2- 

3m  — 4     3(w  +  w)  a  +  i  +  c     a  — ft  — c  +  x 

4m  — 3'       m  —  n  '   a—b-\-c^    x  —  a-{-b  —  c 

3  x  —  b  m^  +  n^         x"*"^ 


4m*(l— x)'    x+c"  *         m      *    b(a  +  b) 

7n^n-\-x     X  —  y  x  —  2       5x^4-1 

X         '    x  +  y  '   a  +  4'    9 x»  —  4 X ' 

x—y—m           c  —  d  a  — 2  b     2x^^  —  1 


Exercise  64. 
^-   ^  +  ^^  +  nr3^'   ^^^-2x»-x-3' 

3.   x«  +  3ax  +  3a''+-^^;   a;  +  1  ^  "^  ^ 


X  — 2  a'        '  x*4-ar-12 


4.  (x-y)«;   x  +  m  +  14-^^— -^' 

X  -|-  n 

X  —  10 

5.  3x«-4x  +  5  +  ,f— i^;   x'^  +  y*-. 


44  ANSWERS  TO  THE 


Exercise  65. 


2m  IIP'        ^    2(^H1).     2m«        w^ 

'      ic  +  1    '  m^  +  ^^'  2^  —  2/ 


2.  .      ;     — ii— : : ;;•  5.    — -\ 


6. 


a  +  ic' 

m  —  n^        a- 

-d 

m« 

2y« 

7/1+^  = 

'    a:Ha;y  +  / 

a 

2 

mn^ 

m  —  1 

3.2.  y2„ 

7.    ^i 7-  •  8. 


(m  +  w)2'         (m  -  /i)^ 
2/' (3  a;^ -  +  //") 


^,rn  ^  ^,«y.  ^  2/'^»  "•  ic'"  +  2/" 


Exercise  66. 

n  —  m        n  —  m    —(n—m)  n  —  m  h  —  a 

'   h  —  a         a  —  h  ^  —  Q)  —  ay       —(b  —  a)^        x  —  m—n 

b  —  a  —  (b  —  a)  b  —  a 

,  +-7 -x;  etc. 


2. 


m+ii—x^       —{x  —  m—nY       —(x—m—n) 
m  —  a         m  -\-  a  —  x     n  —  a  —  b 


n  —  b^        m  —  b  -\-  y  ''    m  —  b  -\-  a 

m  —  x  m  —  X  (m  —  a)(m  —  b) 

n  —y  '    (jn  —  y)  (n  —  z)"*        (m  —  c)(n  —  x)  (m  —  y) 

2a:  — y  — 3  a  —  c  +  3 

4.  ^  ■ 


5. 


{a  +  b')(a  —  m)  (b  —  2  x)"*    (a  —  c)  (m  —  n)  (x  —  y) 

ab(m  —  x) 
mnxy  (a  —  b)  (a  —  c)  (b  —  c) 

—^y{^  —  y) 

abc{a  —  b){a  —  c)  (m  —  n)  {x  —  y)  {y  —  ;*;) 


ELEMENTS  OF  ALGEBRA.  45 

Exercise  67. 

amny     bm*y     bmnx      ab^y  ^       c        2  b       5  a 
bmnybmny^bmny    bmny      abc^  abc*  abc 

cm-\-cn    am— an    bn  .   Sa^bm  3a^m    Sd'b  —  Sabn 

abc     *       abc     ^  abc    3abm  Sabm         Sabm 
bmn 
Sabm 

10 m n-\-20n^     10  m^  — 15  mn     15 m  —  3n 
30  m  7*        '  30  mn         '       30  mn     ' 

a;* -4  (x-^l)(x^-^)     (g  -  1)  (a*- 4) 

♦•    (x2-l)(x^-4)'  (x^-l)(x^-^)'  (a;«-l)(x^-4)' 

(m  —  7iY     (m  +  2n)  {m  +  n)         m'  (q -- 6)  (a  ^-f  ^^ 

2  (g  +  ^)  (<^'  +  ^')     4  (g*  -  6^ 
a* -6*  *      a* -6*     * 

120  m  -f  30       10  n  — 5  9m-6 

*•    16  (w  -  2)  '  15  (m  -  2)  '  15  (m  -  2)  ' 

7  2a«y  m«  -  n« 

2  a;  (x  —  y)  (wt  +  n)  '  2  a;  (x  —  y)  (m  +  »)  * 

m  (m'  —  m  X  -f  x^)  n  a  (m  +  a;) 

m«  +  x«  '  m«  +  x« '    m»  +  x«   * 

•  a^  +  xV  +  y*"   a;*  +  x»y«  +  y*'  a:*  +  x«y^  +  y** 

10    ^  (^  •-  y)*    5(^'  +  y')     5y(x^-K.xy4-y')        «' -  y* 

•  5(x»-2^*  5(x»-y«)'         5(x»-y«)        '  5  (x»  -  y»)  ' 

X"  -i-  //"       x^yCx'^-y*")     {x^^  -t-  y«")« 


46  ANSWERS  TO  THE 

a{b  —  a  +  x)  (x  —  a  —  b) 

(a  -\-  b  +  x)  (a  —  b  -\-  x)  (b  —  a  -\-  x)  (x  —  a  —  b)' 

b  (a  +  X  —  b)  (x  —  a  —  b) 
(a  +  b  +  x)(a  —  b  +  x)  (b  —  a  -\-  x)  (x  —  a  —  b)^ 

x  (a  +  X  —  b)  (x  —  a  —  b) 
(a  -\-  b  -\-  x)  (a  —  b  -\-  x)  {b  —  a  +  x)  {x  —  a  —  b)' 

—  a  —b  c 


13. 


(a  -c)(b--c)'  (a  -  c)  (b-c)'  (a-  c)  (b  -  c) 


x—1  6—2x 

15. 


(a,  _  1)  (a;  _  2)  (a;  -  3)  '    (x  -  1)  (a;  -  2)  (a:  -  3)  ' 

9-3a: 4 a; -12 

(a._l)(a;_2)(a;-3)'  (a;  -  1)  (a;  -  2)  (a;  ~  3)  * 

mx  —  am  x^  —  nx 

{a  —  x){m  —  X)  (n  —  x)     («  —  x)  (m  —  x)  (n  —  x) 
ax  —  am 


17. 


(a  —  x)  (m  ~  x)  (n  —  x) 

a;2-2a;-3  -  (2 -f  a;) 


(1-a;)  (2-a;)  (3-a;)  (5-a:)  '  (1-x)  {2-x)  (S-x)  (5-x) 

-(a; +  3)  (a; -3)3  (x'  -  4)  (a;  -  3)^ 

^^     (x^  _  4)  (x^  _  9)  (a;  -  3)  '  (a;2  -  4)  {x^  -  9)  (a;  -  3)  ' 

•-(■T^-16)(a;  +  3)  2  (a;  -  2)  (x^  -  9) 

(a:2-4)(a;«-9>(a;-3)'  (a;«  -  4) >«  -  9)  (a;  -  3)  ' 

jB»"»4.3af"  a;*"*  —  1  a;^"*  —  1 

^^'    (a:*'«-l)(a;«'«+3)'  (a;*'»-l)(a;^'"  +  3)  '  (x*'"-l)(a;'^'"+3)  * 


ELEMENTS  OF  ALGEBRA.  47 

Exercise  68. 

6a»-16a-15  .    IS b^ c +1S b c^-\- 9 a^c-\- 9 ac^-SaH+ Sab* 
^'  36  a         '  72  abc 

12  g»  4- 28 a;' -27.    x*  +  y*  Aa  +  b 

^'  Sx*  '      xV    '  3b     ' 

cp-\-bm  —  an     Sam  ■\- 2han -\- Ibbn 
abc  12  an 

9  a  +  a'  +  12  .      6  (n  —  m)  ^^  a  —  b 

4.   n >  rrz       '  6. 


7. 


3an  triTi  n 

a«-3a6c  +  6»  +  c*.    a^' +  a'c'-^*c« 
a  6  c  a^h^  c^ 

2a^  2w^ 

13. 


8. 

4  n  —  w  — 

^='  +  «^ 

mnx 

9. 

31         47 

19 

16x  '  42m 

30  y 

n 

a«6  +  ft«c- 

•  ac* 

14. 


11 


ar*  —  y^ '    a;*  —  m*a5 

m  +  4  -2 

m  —  4 '   4  ?«,*  —  m 

m4- w,      2ax 
abc  '^"'   m  —  n     8a;'  — a* 

1  .       Sx  m  —  7 

'   ar»- 9a;  4- 20'   ^^^^'    4m(m«-3m  +  2)* 

4a  — 66  5 

"•    3(m«-n«)'    (a;-2)«(aj  +  3)* 

16  0-     ^^^ 

'     {m  -{-  n  +  x)  {m  -\-  n  --  x)  (m  —  n  —  x)' 

17  ^^y*  .         1  -^        2m^ 
•'^-   a;*-/'    x»-y*'              ^'*   m«  +  n»* 

g'+  7  2  m*  — 2ma;« 

**•   ar«  +  6x  +  8*  ^-      {m^^x^Y  ' 


48  ANSWERS  TO  THE 


21. 

a; 

27. 

^'  +  ^'      ;  2y. 

l-a;« 

22. 

96  x^ 

28. 

1 

(3  +  2  x)  (3  -  2  x)^ 

X-1 

23. 

0. 

3^2  _  ^2  _  ^2  _  ^2 

1 

29. 

(^  -  a)  (X  -  b)  {X  -  0) 

24. 
25. 

X  +  y 
-1. 

30. 

c  —  a  —  b 
(a-c)(b-c)' 

26. 

1 

31. 

2x»    .    Sa  +  x 

2x  +  l 

x^  —  4  '      a  +  X 

32. 

1 

0; 

2x2- 
64 

9a; +  44            33^   ^^ 

(rn^x){x^2)' 

+  x« 

Exercise  69. 

1. 

J3   ^      3.m  +  « 

8. 

m  {m  —  n). 

'  10'    2/'"  +  "' 

2. 

8       4cie* 

9. 

3. 

1          .    3a^ 

-1 

-2 

10. 

x^      a^      y"-       b"" 
a^"^  x"       V      2/'' 

2  -  X  -  ic^ '     X  - 

4. 

aj+l .              ^ 

11. 

(a  _  c)2  -  6^ 

x  +  5'    rr^  —  mn 

4-71^ 

a6c 

5. 

12. 

1 

6. 

m 

13. 

a;«-l 

m  —  n 

a:«+l 

7. 

m^  —  n^    .    X  + 

14. 

^2n  _  y2m 

2  (a;2»  +  2/''-) 

ELEMENTS  OF  ALGEBRA.  49 


Exercise  70. 

acmx     3  4  xy   .         xy 

3(a-^b)\  a*-^2x^y-\-2xy'  +  y' 

b{a-{-b)'    x*^3z»y-^4x^y^-Sxy^-{-^' 

x  +  y        .         1  ^     b  —  X 


m*— 2m4-4     sb*— y"  *   a  —  x 

2  a;  — 1  g  +  6^  —  c 

**2x  —  3*  '  a-\-  c  —  b' 

"•      »*  -  n*x^'+x*    '  a^  +  a-'  +  l 
12.  2a;V-4a;V  +  2xV. 


Exercise  71. 
a^-b^  10.    1.  13.    1. 


a  —  12  >!  «rfi+i 


(J  a:  -  y)«  ^  ,   a  "*  « 

3.    i ^lJ-  .         4.    X  +  6. 

x-y  1       1 

5     (a  +  ft)  (g  -  2  y)  X       x* 

•     (a  -I-  1)  (X  +  y)   '  „       J 

2a6  b       a 

7.    1.  8.    1.  9.    1.  1^-    ^'-2+  -,. 

^   m5    77tt-r,r-^;  8^^-«- 17     ^^      .  18.  lOa. 

h^      J8  6-y"  ■*-^-   aP^c* 


50  ANSWERS  TO  THE 

Exercise  72. 

03  +  6,    a  ■}-  ni  ^    an  -{-  cnx  ^     mnot^y  —  3ic^ 
ic  —  6  '    h  —  m^    cm  +  cnx^    m^  n^  •\-  2  in  n^  x 

h  ^    m  -\-  n  ^     7ri^  —  h  n  ^       2  m 
a  '    m  —  n      a  7i  ■\-  b  in      m  —  n 

x^  +  1       x'^-U  1 

3.     —^ ;      -t; T^  .  5. 


2x     '    x^-lU  -  2j(rn-n-p)'    2x^-1 

..     .     r     (a+x)(a'-x')  x'+l  .  -,     x'-x+l 

1  +  a;2     a;2  _  3  ^  ^  1  4 


7. 


8. 


14-a:  '    ic^_4£c  +  1'    3(1  4- a;) 

*  +  2/ .  am-{-  adn       _    £C  —  2/ 

y      '    bni  +  cn-\-bdn      x  +  y 


m  -\-  n  wn 

9.    a  +  a;;    1.  10. -2-  11. • 

{m  —  n)^  m  -\-  n 

Exercise  73. 

^'   ~8F'    16x''y^''  x'2/     '    256x12* 

160^"^^.    (a;  +  yY  .    rn'ix-ijY 
^'    Slm^nf^'    {x-yY'    n'ix-^-yy' 

{x^  -  i/y  .        (a  -  bY    .    a''(a''-15a^+75a-125) 
*•    (m  +  nY  '    (2c^  +  3  6)2'  ic^^ 

(«_5)2      8  a;'^  -  36  a;^y"  +  54  g;B  y^»  -  275/»"  .     /^y'+" 

5.  .    .  ,..;  Tin  »  I  ; 


_        r   ai  m^  x^      1 ' 


ELEMENTS  OF  ALGEBRA.  61 


Exercise 

74. 

x^       Sm*n* 

6ac» 

«* 

m 

1. 

^^*'           X        ' 

7x» 

'    36*-* 

2. 

a»»    '         b^  ' 

4m?i^ 

5aH* 

3. 

L*a*a''-        a~*  . 

aixi 

5. 

x  +  1-^;    a« 

-hi 

3i/-z'     V^' 

4. 

aj  +  y      .    / 

.^7             * 

6. 

a           ft.    3 

-r*'- 

2V(a;-y)-'    ^ 

Exercise 

75. 

1. 

1       .    x-S 

1  +  x^'    x  +  S' 

5. 

46;  g. 

2. 

Ty^  —  Axz 

6. 

]§.         8.  0. 

10.  -H 

6««-7xy* 

*;   0.    9.  8. 
a 

Sx*y-2» 

7. 

11.  -f 

3. 

9  x«  -  y««  * 

15. 

(a  -  cy  -  b\ 

17.  1. 

4. 

aj  +  5 

16. 

a:2  4-  3  x  +  3  - 

12. 

":+"%.»; 

^:-i 

+  K 

13.    m«  +  —  4-  ^  +  i;    :l  _  ::^  +  !^^Jf  _  '^, 
n^  n"^  v}'    y'       ny"^  n'y       n» 


52  ANSWERS  TO  THE 


18. 


19. 


\n       inl  \n       inf       \  a/  \  a^J       \ic**       y^l 

(a^       ax      x'^\ 
l?~b^'^  yV  '■ 

5^(y'"  +  l)(y"  +  l)(2/"  +  l)(y"-l),orx*»(2/^"+^) 


20.    ic.  ^ "~  y  ''®-  ^- 

2^    28  (a: +  4)  ^^'    ^2/    *  3^  ^ 

36.    ^-±^  .  41.  3. 

^c  +  ac  -*-  a6'  42.  1. 

2b  (by  —  ax) 


22. 


23. 


(«,_,)(5_c) 
24.-^. 

25.  1.  28.   0.  30.   1. 

26.  2.  29.  a;.  33.  3. 

27.  0;   (a«-68)2.      34    2. 

a:(a^  +  l)     .  1.  1  45.    '^ 

^^'  x'  +  ^x  +  r  '  («^  -  ^0  (^  -  ^) 

32.    '^;    ««^'^c^  46.    4- 


o#. 

ax  (S  ax  —  5  by) 

40. 

16  aH^ 

43. 

2{a  +  b  +  c)^  +  a''  +  b''  +  c'', 

44. 

1 

c  (a  —  c)  (b  —  c) 

^1; 

2  a 

ELEMKNTS  OF  ALGEBRA.  58 

Exercise  76. 

1.  10;  f.  4.   6;  -|.  6.   2.  8.   4. 

2.  20;  5.  5.    i.  7.    2;  6.  9.   -2;  14. 

3.  2. 

Exercise  77. 

1.  1;   6.  5.  8;   0.  a  4.  11.  3. 

2.  1.3;  -2.  6.  -4.  9.  4.  12.  -.04;  i. 

3.  0;    f  7.  1^.  10.  %.  13.  li;  3. 

4.  2;    0. 

Exercise  78. 

1.    n — ;— ;    wv(l-3a).  5.    a-b;    — — r 

2a       2ah  ^    n  * 

^*    36  '    a  +  ft  <? 

1  ft       I         ^ 

3.  4m  +  8n;   -T-  8.   0, -;   ft-1. 

'aft  c 

a  •  w» 

4.  j^.  9.    i^  +  27^- 

Exercise  79. 
^        c    ,         ,,  13.    H«  +  ft  +  3). 

3ft  2a» 

2.   — •  3.  4.       ^^    28«+5ft«*  12.  5. 

5    ^(a-6+c).         *•  ^*       16.  —mn  +  mp-{'np.l^.  0. 


6.   0,-;  17a.         7.  3(n-l). 


11.  2. 


a 
a*  2  oh 


o 


b  —  a*    a  +  ft 


6.  -  "^ .  3  rf»  _  2  c«  17.  36. 

21      19.    — To— TJ— • 
12  c  a 

7.  0.  •         18-  ^• 


1  n  (n^  4-  m') 


54  ANSWERS  TO  THE 

Exercise  80. 
2.   31.  3.   84.  4.   36.  6.   1^^  days. 

abc 


ab  -{-  be  +  ac 


days.  8.   10  days. 


2abc  .  2abc        .  ^         2abc 

9-    —r-, 7-T-'     -A.,  —^ 7  days;   B,      ,   ,  , 

ab-\-  ac  +  bc  ac  +  bc  —  ab      "^  ab  +  bc  —  ac 

2abc 

days;   C,  -^j—, r-  davs. 

-^    '      ^  ab  -\-ac  —  bc 

10.   48  minutes.  12.    16  miles.  13.    ^       miles. 

b-{-G 

15.  150  miles.  14.   742^  miles. 

16.  56  hours;  84  and  70  miles,  respectively. 

acn      .  .        acm         ..         _,        abn 

17.  - — ; hours:   A,  - — ; miles;   B, miles. 

bn  +  cm  bn-{-cm  bn-\-cm 


18.  First  kind,  — ^^ ;    second,  — ^^ • 

m  —  n  m  —  n 

19.  ^^.  21.    69.  24.    160  miles. 


2  m  n  {2  m  -\-  n) 
4:  m^  ■\-  4:  m  n  —  n^ 


20.   -, — 2—-^, =^-^  days. 


22.  $1200  in  5  per  cents;  |2000  in  6  per  cents ;  sum,  $3200. 

$100  ^•m         .  ^100  b(n-m)    .       ^ 

23.    ^ in  a%; — ^ ^  in  c%; 

am  -\-  en  —  cm  am  -{■  en  —  cm 

$100  b  n 
sum, 


am,  •\-  en  —  cm. 


ab          -1            ,                       a(3w2_^3w  +  l) 
25.    miles  an  hour.       26.   — ^^ ^!^ — — - . 


b  —  ac  (ti  +  1) 


ELEMENTS  OF  ALGEBRA.  55 

Exercise  81. 


(a:  =  24,  I  «  =  H 

i3,  =  12.  "•   13,=    7. 


3.  }^  =  2J  a.  J^  =  24.  12.  j^  =  l^ 

Exercise  82. 

*•  ■jy  =  3.  "•    (y  =  12.  "    U  =  3. 

_    («  =  3,  (a;  =  38J,  ja;  =  l, 

'■  ly  =  2.  1y  =  -21J.         •   (y  =  4. 

6? 
SI 

1. 


'•iy  =  60.  *•  iy  =  -6.         "•iy  =  S 

ty  =  12.  '•   |y  =  7.  ly  =  - 

,x  =  19, 


r  X  =  i», 

1y  =  -i- 


1. 


Exercise  83. 
x  =  -t,  .    (1  =  21,  „    f«  =  13. 


)y  =  -i.  1y  =  i5.  ty  =  ll 

a.  J''  =  J'  c-r^fi  10.  1''  =  ^' 
(y  =  2.                 (y  =  6.  )y  =  5. 

fx  =  -3,  <x  =  10,  11    j*  =  4' 

'•iy=12.  '1^  =  11.  "•1.'/  =  6. 

4    (*  =  *•  8    i*=    ^'  12    i'  =  216, 

'••ly  =  -2.  "•ly  =  12.  ''•ty  =  144. 


56  ANSWERS  TO   THE 
Exercise  84. 

^•iy  =  .02.  "•   1^  =  12.  '^-  1^  =  12. 

3.   .P  =  -2'  15.   r  =  12,  27.    j^  =  8, 
b  =  -3.                 |y=    6.  (^  =  7. 

fa,=  10,  ra;  =  7,  p  =  4^, 

ly=    8.  U  =  5.  ^"-  t2/  =  5i. 

5.  5=^  =  25'  17.  1^  =  10'  29.  r  =  f 

«-L=4:  ^°-{,=i5:  ='='-i,=io. 

^•{,  =  12.  ^^•{,  =  1.  ^^■],  =  .08. 

f*  =  ^'  22    jx  =  20,  (a;  =  i, 

iy  =  2.  ""•  1^  =  21.  ^*-iy  =  2. 

jx  =  60,  23.  r  =  2'  35.  r^^ 


10 


11 


12. 1^=^'     24.  r=^' 

1  y  =  4.  *  y  =  7. 


ELEMENTS  OF  ALGEBRA. 


67 


X  = 


2.   r 

(y 
(y 

X 

y 

X 

y 

X 

y 

X 


\y  = 


6. 


7. 


(y  = 

rx  = 

\y  = 


X  = 


y  = 


i 
§. 
-1, 

-2. 
-44, 

6, 

8. 

12, 
6. 

am—bn 


Exercise  85. 
9.   r  =  ^' 


15. 


x  =  h 


11 


12 


13 


<x  =  . 

\y  =  l 

{^  =  h 

'  \y  =  l 

(x  =  -l 

iy  =  h 
(x  =  ie, 
\y=  7. 

<^  =  h 
\y  =  h 

Exercise  86. 


16.   i^  =  J' 

(x  =  i, 


17 


18 


19. 


20. 


21. 


x=l, 
1. 


(y  = 


«  =  iWr, 


J" 

I- 


x  = 


y  = 


ar  = 


y  = 


an  —  bm 

a^-b^    ' 

nq  —  mr 

I         a*  +  aft  +  &• 

Iq  —  mp  ' 

6.  - 

^-        a  +  6 

Ir  ^  np 
Iq  —  mp' 

aft 

l^=      a  +  ft- 

be 

7.  - 

\x=''^p:. 

qr-p^ 

ac 

.,    -P9'-*** 

a«4-ft»' 

l^-,v-y 

«  +  fli 

ax^  —  a 

OiA  +  afti ' 

8.  i 

X  =  Sf 

Oi  —  a* 

*,-6 

1  —  a  Ox 
1/  — *. 

axb  •\-  abi 


Ox  —  a* 


58 


ANSWERS  TO  THE 


9. 


10. 


11. 


12. 


13.  < 


14.   < 


am^n 


h 

+  b 
a 


a  —  b 

,2 


a' 
0  {a  —  c) 


y  =  - 


a  —  c 


nr 
X  —  — , 
c 


X  = 


c{a  +  b) 

2^ 
c  (a  —  b) 

2a 

2n 
m  -\-  n 

2m 
m  ■\-  n 


15. 


f  m  — n 


X  = 


m-\-  n 


aai  (a  —  ai) 


16.  ^  "  -r  «i 

I      _  aai(a  +  ai) 


17. 
18. 

19. 

20. 

21. 
22. 

23. 


r 

\y 


X  =  m  -{-  Uf 
m  +  n. 


a 
be 


I- 


a  +  2b 


x  =  ~  } 
m 


n 

1 

—     > 
n 

L^       m 
r  a;  =  a  -|-  5, 
\y  =  a~b. 
(x  =  m  +  nf 
\y  =  m  ~-  n. 

a'^jb-  a) 
^-     a«  +  6»     ' 


y  = 


a^  +  2b^-ab 
b  —  a 


24.   { 


X  =  zfy 

mn  —  1 


y  = 


4-1 


mn 


25. 


(  X  :=  m  —  Jlf 

\y  =  n  —  m. 


ELEMENTS  OF  ALGEBRA. 


59 


Exercise  87. 

( 

x  =  3, 

x  =  l, 

1 

x  =  -, 
a 

"1 

( 

y=2, 

.z=b, 
x  =  10, 

4.  ^ 
f 

y  =  2, 

z  =  3. 
x  =  -5, 

7.  < 

1 

1 

"• 

y=  2, 

^• 

y  =  5, 

'x  =  20, 

1 

.2=3. 

I 

«  =5. 

8.   < 

. «  =  30. 

ar  =  2, 

( 

-:r=    7, 

rx  =  -12, 

'■{ 

y  =  3, 

.«  =  4. 

M 

.;s=    9. 

9.  ^ 

y=  6, 

U  =  18. 

a^i 

5 

> 

2 

^-a6  +  *c 

4-ac 

~  ?/l  +  71 

10.  < 

abi 

? 

9 

12.  < 

2 

^~a6  +  6c 

4-  ac 

^      m-j-  p 

abt 

5 

• 

2 

[         a4  +  ic 

+  ac 

.     ~  w  +/>" 

par  =  4, 

rx  =  -li, 

r'  =  i, 

14.   . 

y  =  5, 

17.  ^ 

;y  =  2i, 

11.  . 

[v  =  3. 
\x  =  a, 

U  =  6^. 

a 

15.. 

y  =  b, 

18.  . 

y  =  -3i, 

»  =  2' 

z  =  c. 

U  =  2^. 

13.  V 

/=2- 

16.  . 

rx  =  l, 

19.  i 

fx  =  7, 

y=9, 

U  =  3. 

60 


ANSWERS  TO  THE 


2. 


3. 


iy  = 


3, 

2. 

13, 
5. 


y  = 


a  +  b 
0. 


4. 


r 


z  = 


a, 
b. 


6. 


Exercise  88. 

(x  =  2, 
13/ =  3. 

^  =  ^^^475-2' 
a^bc 

\  X  =  5^7^, 

^  ~  abi  +  aib 

2  a  aib 
^  ~~  abi  +  aib 


11. 


10. 


mn 


12. 


13. 


14. 


^  —  ^^ 


2/ 
U 


—   3  5- 


y  = 


m 


X  = 


15.  ^  2/  = 


z  = 


f._ 


m 

+  n- 

Sp 

w 

a 

—  n 

n 

b 
+  3p- 

—  m 

m 

c 

20. 


f      _  (a  +  ^)  w  + 

_  (a  +  ^>)  ^  + 


2, 

3, 
4, 
5. 


^m 
"^X" 


16. 


VJ.\y  = 


2  m    ' 
mn  -\-  mp  -{■  n  p  —  n^  —  2  p"^ 


z  = 


2(n' 
3np  —  mp  - 


p') 


m  n 


2  (n^  -  P^) 


19. 


y  = 

z  = 


18.    f  = 


ELEMENTS  OF  ALGEBRA. 


61 


X  = 


21.    ^  V  = 


mnp 


mn  +  np  —  mp 


2mnp 


z  = 


nip  +  np  —  mn 

2mnp 
VI  n  -f  itip  —  np 

ar  =  ^a  +  *  +  c)  —  a, 


32. 


X 

= 

2 

mnp 

mn  -f 

np- 

mp 

y 

— 

2 

mnp 

mp  + 

np- 

mn 

2 

mnp 

mn  -\-  mp  —  np 

1 


x  =  -  t 
a 


23 


rar=|f  (a  +  ft  +  c)-a,  i 

I  z  =  §  (a  4-  *  +  c)  -  c.  1 

27.    f^  =  ^^*  + 


W*  -I-  7i*, 
>8 


30. 


31. 


34. 


X  =  105, 
y  =  210, 
z  =420. 
a;  =  l, 

y  =  2, 

«  =  3. 

x=  7, 
y  =  10, 
2=    3- 

=   7W,*  


35     (x  =  m''-n", 


2.    J. 
3-    h  h 

4.    ^. 


Exercise  89. 
5-    A- 


6. 


a-\-bm  b  +  an 
mn— l'  mn— 1 
10,  -1. 


7.    16i,  15. 

cm  —  an 


10.   Corn,  i;  oats,  f . 


,,     /t  bn  —  dm 

11.  Corn, ;   oats,    . 

oc  —  ad  'be  —  ad 

12.  180  lit  2  for  3  cents,  300  at  5  for  8  cents. 


62  ANSWERS  TO  THE 

100  a  (b  c  d  -  12  en  -  12  np) 

.  777 r at  a  eggs  for  m  cents, 

13.  ^  d(b  m  —  a  n)  °°  ^ 

100b(127nc  +  12pm  — acd) 

TT, — c at  b  eggs  for  n  cents. 

a  {b  m  —  an)  °° 

15.  245.  ^l^L±_?!l!!) 

^^*   "  2  a  -  81      ■  20.    853. 

16.  891.  ,  ,, , 

19    ^^^  +  M11-^),  21.    315. 

18.   39.  11  — a  — c 

23.  A,  120 ;    B,  80  ;    C,  40 ;    altogether,  21^9^. 

.    ac(n  —  m)     ^.    ac(n  —  m) 

24.  A, ^^ ;    B,  ^ . 

nc  —  a  mc  —  a 

25.  Arithmetics,  54 ;   algebras,  36. 

26.  Crowns,  21 ;    guineas,  63. 

,.,  nia^c  —  a  Ci)  .  n(aic  —  a  Ci) 

27.  Crowns,  — ^ :   guineas,  — ^^-^ ^  . 

a^n  —  am  cm  —  c^n 

28.  A,  50;   B,  21f.  30.    Stream,  2;   A,  10. 

.       mn       ^  mn 

29.  A,  ;   B, 


31.    Stream, 


a  a  -\-  n  —  m 

mhi  —  rrixh      .     mhi  -\-  m^  h 


2hlH  '^hK 

32.   39  miles,  8  miles  an  hour.  33.   5. 

34.  Going,  4^;   coming,  1\\   stream,  1\. 

hh  .  ah  ^  m  (a"  —  b"-) 

35.  Gomg,  ^-p-^;     coming,  ^-^;     stream,        \^^^   ^ 

m{a-\-  by 

2abh      '  36.    Sugar,  5;   tea,  60. 

r^  100  nOO  (71  -  m)  -hm'] 

3,J^-^-' TW^) ^ 

100  [am -100  (yi-m)] 
^ea,  ^^  (a  -  6) 


ELEMENTS  OF  ALGEBRA.  63 

38.  Sum,  $1000;   rate  per  cent,  6. 

39.  Sum,  dollars;   rate,  -— r— • 

40.  First  kind,  14;   second,  15;   third,  25. 

41.  Fore-wheel,  4;   hind- wheel,  5. 

42.  Fore-wheel,  ; (rnr-'sn) ^  ^^^^     hind-wheel, 

'  (m  +  7i)  (cs  +  cr  -{•  ar) 

b(mr-8n)  ^^^^ 

\$  +  r)  {em  +  en  -\-  an) 

„.    ,  .  .    .   bnti  —  bim  aim  —  ami 

43.  First  kind,  — { \— ;   second,  — r j-  . 

aib  —  abi  Oib  —  abi 

44.  First,  3} ;  second,  3. 

^.    ^    a(m-\-l)  .    a  (n -\- 1) 

45.  First,  — ^^ T^;   second,  — ^ :r  * 

mn  —  1  mn  —  1 

46.  3^8,  80%  ;   4's,  125%. 

bd  -\-  np  bd  —  mp 

4a   A,  55;  B,  105. 

A    ^{n  —  b)  (m  -\-  n  —  a)  c  {a  —  n)  (m  -\-  n  —  b) 

*^'      '  m~(^r:^)  '   ^»  7^7^^^6) 

50.  Sum,  500;  rate,  .04.  * 

.,     o         bm  —  an        ^        n^  m 

51.  Sum,  —7 ;   rate, 


b  —  a  bill  —  an 

52.  Pounds,  60;  cost,  28. 

53.  Larger,  5.678;   smaller,  1.234. 

54.  Smaller,  :j -;   larger, - 

1  —  no  1  —  a  0 


64  ANSWERS  TO  THE 


1.    n^^   i. 

216  a«         1 
3. 


^         a.^  +  1 

4       ™n  +  I  .     1   .    . 


Exercise  90. 

0    a' 4-   /^«'-*'Y 

"•''*'  u  +  w 

0    ^-          ^ 

2"  '  a'  (a'"  -  b*") 

11.     ^"'f'.^  +  lZ;^'''-"*-!; 

1 

1        16 

16"  a"""*  a;""'  y^ 


'  •  m    m  >        1  27        •     


Exercise  91. 


13.  aVa;^  6^2  6^c2«'. 

14.  2".  15.   22'". 
16.    (a;2-y2^-3«j  2«. 


1.    ^  ;  1 ;   ^a;^  1.  8.    V-{a-2  x) 


9  ^\17(l  +  a) 


a»'  _i_.   L_.  1        9.    a;2«;  1. 

1         -     „-    1      10-  (^  +  2/r;  ^"-^'^  -6 


3-   ^r-;;r;  va»;  Vet'";  - 


«- 


11. 


_1_         /^\{m  +  n)^ 


4.   —  4  a/2 ;   ^a\  xy. 

6.   3";  (a -5)"-;  g-  13.    ^a ;  2«'. 

7     ^\'/"^.  ^'  14     a-  a;«  +  ^+-^"- 


ELEMENTS  OF  ALGEBRA.  66 

Exercise  92. 

1.  9;    X;  4m-*;    -^;    ^. 

2.  IG;    o^^i'^;    516a»''6»»-^c»''-«. 

3.  343  aV"*"*;  (27)'»6»'-«  Va*"';   a:*- 

5.    |,;   y»^-^-;    25.  ^-    S '    1 5    V^' 

7.    _^L^j    (25)'"'»a'''r***'"c'^'"^"S    J^4.       8.    a:"'";   a^'^"- 


Exercise  93. 


1  ,1  ,1  (b-\ 

^-  (^-Y  <""">'  (^»'  (''^^''  (;^'  i^'i 

Va-'ft-'y-*'' 

_1_    «    m   

4.    -fo^;   ^iO^^  yaHW',    yJ  a^^  b" ;  \  ab^-,  ^  x^  i/\ 

6.    Sa^bc^',  5a-^b   hr^;  mlnij  ab^;  x-^t/-^;  ^"*y^;  «'**• 

2         •     1  1  1  5  _  1 

^-   5a-«ca:^   a-J^!^    -^^^   xjri^*^   Sa^J^V^   ^""^' 

rt*  1 

8.  «*;  jgj    «*"•;   a»;    -;    al. 


66  ANSWERS  TO  THE 


m 


10.   -;    7^TT-^',   1-  20.   -t;   1;   1. 


12.    |;;16.^m;^;  22.    2«%^ 


M     ,  ^^  23.    I;    18X10^^' 

\/nr  J/7.  ^"^       3' 


2  »i  m       t  t 


1  6/- 

a;  " 

Veto     *  "?         ^   ±         1 

27.    a'*  —  a2»^»4«  ^  523 


a^ 


Z.2        ^  29.     £c2n-2_      2m- 

17.    ao'';    fw7^ic.  "^ 


t  m       t 


11  .  . 

32.    a**  —  1  +  a~'* ;   n^ y^  —  n^m^'xiy^  +  mTn*xiy^  —  m^n^x^y 
+  m^n^x^y^  —  m^nrxy^  +  ?7i7a3s. 

(as"  +  &-'*)(ai'*  —  b-''). 

b^      Ab^x        6a;2        Ax        x^       ,        ^  12        8 

34. ^  +  — T 72  +  745    x^  —  6x  +  —  +   -^', 

a*  a^  a^         ab^         b^  x         x^ 

5n  .  5n 


a       2    _  5  ^-^n  _^  ^Q  ^-^n  _  ^Q  ^^n  _,_  5  ^f , 


a 


35.  2!  X  3^,  2i  X  3^,  21  X  3^,  21;   24. 

36.  2§  X  3^,  2i  X  3^,  2i  X  3^,  2i,  2i  X  3^,  2§  x  3*;   576. 


ELEMENTS  OF   ALGEBRA.  67 


37. 

ai  +  i;  26i- 

a-H\. 

43. 

a'-'-^+a;  2^^' 

38. 

2a-\-a-\. 

44. 

77»    ^ 

39. 

x-^x-\ 

y' 

1 

45. 

xi;2. 

40. 

X 

,,-^  ,;  2»«%  2. 

X 

46. 

♦1. 

3-2a^-. 

i^m" 

42. 

x-^  -.  1. 

47. 

(5a:^-3yY;  m« - 

-w«. 

Exercise  94. 

1.    V^i;    \^V";  ^64;  ^i*"";   ^8"^;  ^27  aH*c^;  '^/W^', 
^;    ^^;    ^i^^.     y/i;    y/J;     ^/2^;    ^F; 

a.  Vi;  ^i;  V800;  VH;_v/^¥^;  v^^^. 

5.    VW^^l^^,    v^ii^^l;   V/^- 

8.  12  V2;    15\/6;   -9v^;    |  \/58;    \a</Va. 

9.  |V2;   f\^4;   ^-^^20;   7^3;   iV5;iVl5;    i  ^32. 

10.  i'^'24;  VV6;  -3a6^4^;  2aJ6«'^;  ;r^,\/3a*na;. 

11.  a*i-V^c;    -V^;    — V^^aJ. 


68  ANSWERS   TO   THE 

12.    ^  ^6  ;  9  a"  b^"*  \/Wl?  ;   -  ^2ax'y';  x  y""  ^yx\ 


13.    (x  +  y)Vx  —  i/;    (x  —  4:)^a. 


14. 

t>(^  +  y)'''' 

7 

^3n^2,n^5^ 

—  m 

15. 

72;   3;   2. 

20. 

fa^c;    a^oTx. 

16. 
17. 

21. 
22. 

1. 

3     2 

2'    «• 

18. 

10  a  m  X. 
Bab. 

23. 
24. 

2  a. 

19. 

a</4;   2a«Z»'^8a2i» 

Exercise  95. 

1.  v^8;    ^9;   §^1296;   3^64;    ^^»;    ^8;   J  ^16. 

2.  V3;    tv^64;    {^«^';    •'^?. 

3.  -v/a^?";   y/o^^;    y/a^    SJx'yl',    ;/a-;    y/-^;    :.'Y«'- 

4.  v^l25,     V^m,     V^l3;     v^i024,    v'625,     y'MS;    -^^Gi, 

^81,    ^6. 

5.  ^2,  ^2,  ^^2;  ^49,    ^625,  ^2l6;  ^a«,   3^^°. 

6.  ^^,   ^^;    ^a^,   ^^^   ^^;    ^^,   ^^^ 


7.  V<   ^625^,  ^27^;    VW^',  V27 b^   V^^x^ 

8.  a:^<  ^^«"^;    ^<   ^<  ^^,   ^^^^^^^. 

9.  '^a',  W,  ^Z^;   4^625^«,  2  v'8T^^  10  a  ^729^ 


ELEMENTS  OF  ALGEBRA.  69 

Exercise  96. 

1.  2Vi4;    eVH;    6^4.         3.    V^l    v^4;    v/f. 

2.  10  VS;    ^ViO;    ^2.  4.    ^Ti;    1.6;    V^. 

5.  V3,  -^7,  ^4;   8 a/2,  5  a/5,  4V7|. 

6.  4^/7,  S'C^iO,  2V^21;   4v'ii,  2^l3i,  3^. 

7.  3V2,  3^5^,  t>^;  3\/8,  2^/8,  2^21. 

Exercise  97. 

1.  8\/5;   tVlO.  7.  c  (6  c  —  a)  \/2  a  *  c. 

2.  2V3;   6^6;    |  v^2.  8.  mnx\/mn^. 

3.  -a/3;    17^.  9.  2bVS~a. 

4.  20 J  a/3;    fA/15.  10.  -  ^'^^~J\ 

5.  ^a/IO;    0.  11.  f|A/6-^14. 

6.  —  Vva*;    2v5.  12.  — ^^ s ^a/wx. 

Exercise  98. 

1.  18;    18 a/7;   25 a/5.  6.    ^^^486;    v^ll8098. 

2.  2;   14  a/G;  i  >^12.  7.    §  >^ ;   J  a!^  ;  14  >J^. 

3.  9v^288;  10 (3- a/3);  S.     8.    -729; 

,—  6(l-3A/3  +  3ViO). 

4.  2+ A/6  +  2VIO;  120.  ^ 

5.  vTO;  T»B  ^2592;  ^500.       9.   A/ lOO;  n-V^;  12v^2. 


70  ANSWERS  TO  THE 

10.  in  n  ^72li^x\  19.    5 ;  4  \/32. 

11.  m^m^7i^x^^',  6\/^iH\     20.   4;  '^3. 

12.  v'72^;  2«m"a;".  21.    \/^- Vn-,  yVv^lSOOO. 

13.  ^V2;  -12-1  V^.  22.    2;  -v^9^^ 

tt  X 

14.  9ViO-12  ^32  +  6^^3125-8^/10;  ^^4  +  2 -^6  +  v^9. 

15.  4(6VlO-l). 

16.  ^^128  -  ^^2187  -  '^972  +  -v^O  +  v'i  -  V6 ;  12  ^^27. 

17.  -  1 ;   'v/y  -  2  ^^2.  23.    49  a;  -  9  a. 

18.  2 ;  -  7.  24.    1  v^288 ;  |  ^^80 ;  20  \/b. 


Exercise  99. 

1.  2(a/3-V2);  ^5(2^/5+^6);  3^/2-4;  3(VT5-VtO). 

2.  4+V2;   -l^yV^;   _^(3  +  2V6);   t^re. 


x—2'^xy-\-y     '\/xy{?»x-\-^\/xy  —  3y)^    a-\-^d^—x^ 
x  —  y         '  y{x  —  ^y)  '  a; 

4.    2  i«^  -  2  X  V^^^^l  -  1  ;      ^V^(^^-  h){a  -  Vb-) 

a^-b 

5V5  -  VTO  +  5^2  +  v/8     2  a  \/a:'- '"«'-« 


23  '  3  a;  s 

5.  14.14;    7.07;    141.4;    11.18;    .1732;    .2236. 

6.  .707;   -.236;    -1.266||;   1.216§;    .268. 


ELEMENTS  OF  ALGEBRA.                            71 
Exercise  100. 

1.   3V3;   |V2;    '^3.  ^^    axV^-^Vy;  —  ^^^^^. 

3.  3.3;   ^VO;   i  ^486.  "^ ""  

4.  2^3;    ^54;    ^18.  ''•    V  ^^-^^ ^ ;  - ^1944 aV^ 

5.  6^5;   i^96;   2^/2.  13-  ^^'^^^^5   1  4- V^. 
7.   5  V7  -  4  VC  +  2  V5.  ^3    5  -  V6;   ^  '^a^^^ 


.-      le/-^ 


3aa; 


8.    .2;    a^6c;    --^aft*.  ^  .-^^     ^    ,      /^ 

*  16.  1  (4+  V15);  r)+  V<. 

17.    ^x  +  V^^  +  Vy. 


9.  !!Lzl!^;  1^96 


Exercise  101. 

1.    ^^;    V3;   2V3;    V^8;   2  A^^i 


2.  i>^;  i^J^;  V«-^;  2. 

3.  4^;    ^n»;   2^'/2;    V2;    2 15^2. 

4.  V^;   3^;    H;   4«^3;  ^  ^a.' 


5.    8aVft;    ^V^i^^^^;    v'-^;    32a'bc\/2abc, 


6.  3a;  648^/^a:^  2a«;  ^^ 

7.  r,;    ^(«-rr;   1;   ^^^^ 

4 


aC 


72 


ANSWERS  TO  THE 


8.  JV3^;  a'"-V«;  va'';  -4- 


1     X, 


9.    -\^n^—'-';   "''^(aH'^)"^^'-'^^^;    m'n' ^/n\ 
n 

1         8/-r-      0^^  iC*'  V^^^  ^^ 
10.    -,;    Va<o;    — ^g 


11. 


^.;   ^2/(^  +  2/)^    2a^^3. 


27 


12.  x^  +  2;r27/2  +  2/';    t  +  4V6;    11 V2  +  9 a/3. 

13.  2  -  3  ^y  +  3  v^3 ;     4  +  2  V2  +  6  -^4  +   6  '>^32 ; 


m' 


'\/') 


m^ 


3/—      ,         O        6/- 


iiin 


14. 


_^ ^6  —  4a"  +  a^";    v<*^    —  — 


^58m-      16 


15. 


5        10      10       5 

^12  +  a«  +  a^ 


^^  +  ^e  +  ;;r2+:;F+-4  +  l;     «*- +  4  a«- 5  +  6  a^- 6^ 


16.    ^V2m7i  +  |--2^m7i2  +  ^^8m7iH404-  — ^Sm^Ti 


,  32m   3,—,       ^  ,  ...       ---     , 


Q>A:m^x^''. 


Tn^w 


w 


17.    ^  +  1-3^2/. 


19.    ^  +  «^^2/'- 


2  2/^' 


Y2 


18.    1  —  2" ;   3"  —  2\ 
1 


6"/;:;57 


20.   ^^a^--2V;i^-  + V^ 


ELEMENTS  OF  ALGEBRA. 


73 


Exercise  102. 

1.  ^9;   1.2;   9;   1;    i;   |. 

2.  a»v^2;   mv^G;   i^768;   4. 

3.  ^^5^;    ^{/243;   6^3',   4  Vm  (m  +  n). 

4.  -V^^?^^;  i^xTy;  e^^^^v^a. 

5.  ^64,  ^?^,  ^l25;     ^6561,   '^EV^,  i?^15625;    15^2?; 

9^4. 

6.  6^*64,  6^16,  12  ^8,  10-^4;    ^a\  ^6",  ^^\ 

7.  Va,  2V3;   3v^2,  3^2;   2v^9,  3^9. 

8.  iV5,  a/5;    V^5,3a!^5;    Vtr  VlO,  t^^VIO. 

9.  §a^9y^2ftv^96■^^^^'9P;    ^  ^50,  ^  ^50. 

10.  l</%  \^%  ?  ^9,  ^;   2  V5,  ^  V5,  20^/5. 

11.  V2,  V2;   2a;V2,  %bW%  « V2. 

12.  i  ^,  2^6.  i  ^6,  i  ^6,  i  ^6;   2  v^,  ^2. 

13.  4^^,  3  \^3,  3^/275-,   3V19,  5^8,  3^9. 

14.  2  ^8,  5  ^2,  §  V3,  i  V5;    ^H,  Vf 

"•    ^^^VS^»  -^^5^(2i^^,aV(8^^. 

16.  16  V3;    7^7.  20.    8^4;    v^3. 

17.  4>^l6l25;   3 15^4;    ^624288. 

18.  3^/2;  -  ^3.  21.   H\/15;  T^y. 

19.  24v^4;    1.  22.    i;    6. 


74  ANSWERS  TO  THE 


23.  |172;    0.  34.    1;    i;    ^^  ^^. 

24.  -/s'^^;    ly^.  35.     -^^V^;    0. 


25.  156-24V4.  ^^      cm 

_  36.    11;    — -. 

26.  f(7-2VlO);iVl5.  ^"^ 


Vw/^  +  «^  ^^  —  <^  ^     1 


27.  ^l:::_lj^l^! — ^-.  -V^«z/---  _j  V2+iV3+^V3o. 

77t  y 

28.  ^  v^lOSOOOO ;  204.8;  ^a'h'', 

29.  47;  27.  V^.  icB  -  2 

30.  a^-i  +  ljar^y-t^n+D.j/^  ^    '  xl  +  2 

31.  a^^.  39.  |V2. 

32.  cVc;  ^a.  «,y^2o  X  315 


,.y/^ 


33.  c  ^c ;  h  '^Wb,  V        2^^ 

38.  1 ;   Va  (a  V^  +  3  a  +  3  \/a  +  1). 

Exercise  103. 

1.  3V^^;  2\/^;  2V3^xV^;  2aV^;  iV^. 

2.  7a"^»V=^;   3'^=!;    a^^/=l;    2'^-^. 

3.  -1;  -V=^;  -a/=3;  +1. 

4.  +V^;  +1;  -V=^;  -V^=^. 

5.  _4Vin[.  8.    2&(2a2j-l)  V-^. 

6.  4  V^  -  i  V^3  -  3  V^^ ;   0. 

7.  a2(8a  +  a2_^y'ZrX.         9.    -3V2;    -36^/3. 


ELEMENTS  OP  ALGEBRA.  76 

10.  -48;   2,^/^-i,  16.  6»-a» 

11.  ~GV<i  — ^9.  17.  V3j    AV3. 

12.  39-2V^;   21.  18.  i;   2V2-\/3. 

13.  2;    — 4  — 5\/^^-  19.  4;    -  Va  —  V^. 

14.  30  +  12  V2.  20.  1  -  V^;    1  -  V^. 

15.  (a  -  h)  V^^;   a  +  «.  21.  i  A^2625;    V3. 

22.  1  +  V^^;    \  V^^  -  i  a/3. 

6  +  2V^  +  3\/^~a/B     24  +  10V3-15V6-8V2 

^-    3^6 ^  43 5 

lV-1. 

^         / — T-     a*  —  2  a  V—  «  —  a;      2  \/a^  —  a  —  x 

24.     1  —  V —  1 1     7. : } • 


Exercise  104. 

1.  V5-A/2;    V3  +  'v/2;   4\/2-3;    iVB  +  1. 

2.  VI0-2V2;    V6  +  V5;    V7  -  \/6. 

3.  V14-1;   2Vll-V'3;    V^-l;    V6  +  2. 

4.  2-iV3;    V^2  [a/3-1];   a/5[V2  +  1]. 

6.  m  Vw  —  V^  w;    a/'J'  --  a/2. 

7.  m  — n  — 2Vmn;    /v/^('y/2  —  1). 

8.  2-V3;   i(V5+l);    a/5  +  1. 

9.  a/2  +  1;    V5-V3;    V^(V5+V^). 


76  ANSWERS  TO  THE 


Exercise  105. 

1.  11;    10;    -f.  ^^     (x  =  7^-^VTS, 

2.  4;    15.  *    (2/  =  |A/i5. 


c  — 


1       9n^ 
12.    —;    

34        m 


^-  ^^'  "^'^^  f       2V«(V«+ v^) 


iC  = 


a  —  b     ^^  I  "^  I)  —  a 

;  54.  16.  ^ 


I  ^  ^  2(Va6-l) 


5.    2;   -,V  '^  ^-^ 


6.  192;  3^.  ^^     (^  =  -li, 

7.  4. 


8.    1.  f      _  Q^  —  ^ 

18.   ^^-l-n' 

^^'         4m       '^'    ^i-  r  m  +  y.-^ 


- } 


13.    15;    6.  ^^     I  Vm 

14     ^(^  +  ^)  . 


h 


Vn 


Exercise  106. 

1.  0.7781;  1.8060;  1.1461;  0.9030;  1.0791;  1.1761;  1.9242, 

2.  2.5353;    1.2040;    2.3343;    1.4313;   1.6532;    1.5562. 

3.  1.9542;    2.3222;    3.5562;    3.0491;   3.2252. 


ELEMENTS  OF  ALGEBRA.  77 

Exercise  107. 

1.  4;   1;   2;  -2;   -1;   -1;   -3;   -7;   0. 

2.  0.8281^   2.8281;   1.8281;  T.8281;  3.8281;   6.8281. 

3.  "Six;"    "one;"    "four." 

4.  "Fifth;"    "first." 

5.  T.2552;     1.3522;     0.0212;     0.5741;     1.0212;     0.7993; 

2.0970;    2.6232. 

6.  3.7481;    1.1070;     1.1582;     0.0970;     1.0970;     2.6990; 

5.4983. 

7.  T.4804;    0.7323;     5.7781;     3.3222;     0.5441;     4.5441; 

0.6511. 

Exercise  108. 

1.  1.2040;     2.0970;     2.5350;     1.8060;     3.3397;    1.8060; 

1.9084;     1.8572;     6.3588;     1.6110. 

2.  2.5353;   2.5562;   4.1070;   T5.2620 ;  3.5810;  5.2569. 

Exercise  109. 

1.  0.3980;     1.7781;     0.1461;     T.7781  ;    0..5441  ;     5.6320; 

T.3980;     2.4950;     T.5441. 

2.  0.6690;     4.1373;     0.0970;     3.7781;    1.5899;    4.1040; 

0.6990 ;     0.4559. 

Exercise  110. 

1.  0.1690;     0.1590;     0.0602;     T.9466 ;    0.7952;    0.6476; 

T.9964 ;     T.8950. 

2.  0.8063 ;    0.3523 ;    1.2519 ;    0.0691  ;    0..3093 ;    0.5456. 

3.  T.6360;    1.6507;    0.2605;    0.6851;    T.9962. 


78  ANSWERS  TO  THE 


Exercise  111. 


1.  1.8451;     2.0086;     2.3032;     2.9996;     T.8525;     0.5563; 

S.8971 ;     0.5065. 

2.  3.4914;    2.9926;    5.4771;    1.0034;    4.4692;    3.4794.     . 

3.  4.5142;    1.2638;    T.5876;    T.6235;    0.7672. 

Exercise  112. 

1.  2940  ;  .0289  ;  63900  ;  3.151 ;  1.57. 

2.  .00455  ;  30.94  ;  33138  ;  .03333  ;  4.566. 

3.  7.586  ;  50.56  ;  633420  ;  .001301  ;  539375. 

Exercise  113. 

1.  29.77  ;  4.75  ;  2.814  ;  32464. 

2.  2.664  ;  .368  ;  .0769.  12.    9  ;  20  ;  121  ;  30  ;  56. 

3.  373.6;  .4847;  .6186.  13.    2;  30;  G;  14;  1. 

4.  1.683;  2756;  .482.  14.    T.75729. 

5.  9745;  3.264;  3.637;  .4276;  .2163. 

6.  1.53+ ;  9.58+;  6.44-;  5.59+. 

7.  2.56+;  1.2+;  2.  15.    2;  3;  5;  7;  10. 

8.  4.56+  ;  3.46+  ;  3.  16.    -  2  ;  -  3 ;  -  5 ;  -  6 ;  ^. 
log  n    log  n  —  b  log  m  log  n 


9.    -3.3+ 


log  m '  a  log  m         '  a  log  m  -\-  b  log  c 

'09+,        \  X  =  1.177-f 
709+ ;      '1?/=:  .677+ 


^^      |a::::.2.709+,  1  ^  -  1.177+,  17.  g  ;  3 ;  4 ;  6 ;  -  1. 
\y  =  1. 

18.    4;  -3;  3;  6. 
_  4  log  m 

"^-"To^'  (x  =  3,           19.    2;|;i;-4. 

\   _       ^Qg^.  h  =  ^-          20.    -y;  -t;-2;  1; 

^~       log  7^'  -1. 


ELEMENTS  OF  ALGEBRA.  79 

21.  4;    16;    8;    64;   512;   32;   256;    1024;    (2048)\ 

22.  .00001;    5^3;   243. 

23.  3;   i;   .3;   V;  V>  and  t?^;   4,  and  4. 

Exercise  114. 


1.    ±  3;    ±  2\/A-  8.    ±  Vm  +  n;    ±  c. 

a.    ±2J;    ±Vi|.  9.    ±\/^i   ±V^f^' 

3.    ±8;    ±5.5.  1^^ 

10.    ±  V t;    ±4. 

5.  ±.3;    ±tV2.  ''»-gl 

6.  ±2;    ±^.  "•    ±"i±v!!    *•   *^- 

7.  ±|V30;    ±}V77.        12.    ±\/*(6»c-2«;;   ±  J. 

"    c 

Exercise  115. 

1.  12,-2;  8,  -10;  16,  2.       7.    ff,  -  y  ;  0,  4. 

2.  17,-4;    4,-13.  8.    2,-4,-5;    1,-1,-^. 

3.  -3, -i;    i, -3.  9.    2, -V^;    |a,  ^a. 

4.  107,-106;    §,-i.  10     '',-^;    4,-1. 

5.  ^,-2;   2a, -8. 

,       o  11-    1.-^5    ±5,  ±3\/2. 

6.  0,-3;   T^,  J.  'a -6' 

Exercise  116. 

1.  8, 15;  7,  6;  \,-\.  5.   -?,  -i;  V-,  -^3^- 

2.  23,-1;  6, -J/;  1,-4.       6.    V, -§;  A.-?- 

3.  2,-V;  3,1;  3,-1.  7.    1,-3;  i,-J;  ^^-J. 

4.  11,  V ;  h  -4;  f  -A.      8.  V,  -^\  ^i  -f 


80  ANSWERS  TO   THE 

15.  i(5±V22:6);  8,-^. 

16.  11,  2;  7,  2. 

17.  5,  -4f. 

18.  9,-H;  3,-3§§. 

19.  13,  ±VM;  12,-2. 

Exercise  117. 

5.    71)— p\   ±  ^2n  —  ^/n. 

/ — ;-     ha  1 

2.  ±yah\    -,  — T-  6.    a  ±  -;    ^,  —  a. 

a        0  a 

^    .      "^h        b       2b  ^     .  a 

3.  1,  - — -;    -, Q,   Za, —a,  a, —-' 

a  —  b      a         a  2 

a^    __2y      8a    a      5  =b  V25  —  4:m^ 

a(a  +  ^)     a  (a  —  b) 


9. 

3,-t;  -i,-6§. 

10. 

1,-14;   ±9. 

11. 

13,  §;  9,-ff 

12. 

3,  -8.7;  6,  e. 

13. 

2,  ^Y-;  8±  V601. 

14. 

12,-2;  3,-i. 

1. 

,     «        b 

7.     (a^±  ^»)2; 


a  +  b 


9.    (a  +  ^)^  -  {a  -  by;    0,  ^  4-  -  -  c  -  1. 

0  Til 

10     ^  +  ^     <^-^.    a  ±  Va^  +  6'^ 
a  —  b'  a  +  b'  Sa^b^ 

a  f         ^b^  -  4  c''\      l  —  8a±  2aVa 
^^'   2  V  "^     2«^  +  ^    ;  '  (a  -  ly 

13.    ±  1,  ±  ^  V^3;    ±  V2,  ±^V6. 


ELEMENTS  OF  ALGEBRA.  81 

Exercise  118. 

1.  9H,-11;   2,4.  4.   2,  i;   3,-2. 

2.  7,-7|;   -i,-?^  5.    i,-i;   3,  f. 

3.  4, -3S;   2,-8.  6.   4a, -^  a;   2,-22. 

8.  |(-1±V^);  ^,-a,^(-5±Vr4);   ±V;^-1. 
a  -\-  d  ac  +  bm^ 

»•  1' — d-' "' ^ — 

10.  — , ^;    m  —  2  a,  ^  m  -\-  a. 

n  m, 

11.  a  ±  (6  —  c).  16.    2  rt  +  6  *,  a  —  8  ft. 


2  *  w  —  2n'       m  ■\-  2n' 

~^^\  -1  ±  ' 
a  +  ft  ±  a/2  a^  +  2  ft* 


13.    ±  V2  a  -  a^  -1  ±  A/T^=^;    -  (2  ±  Va^  +  4) 

a 


14. 


4,/ _  5ft    a -2ft  2     /J7 

15.       ^     .     ,      »     .    '  18.  ±^V3. 

6aft  3aft  3a 


Exercise  119. 
2.   8,  9.  3.   15,  12.  4.   6,  21.  5.  8,  6. 


7. 


,»iO±\/^)-  ^1^- 


82  ANSWERS  TO  THE 

9.  Greater,  _-«y»L^;   less,  _^V"  _  .  g.   g. 

10.  18.  13.   87.  14.   53,  35. 

11.  Eate,  30  miles  an  hour;  7.       12.  Kate,  6  miles  an  hour. 


2m  —  n  ±.  ^/ii^  +  4  m? 

16.  La]'ger  pipe, ^^ minutes ;    smaller, 

2m^r  n±  V^  +  4  m^     •      ^        t  •       oo      • 
minutes.  Larger  pipe,  88  mm- 

utes;  smaller,  154  minutes. 

17.  B,  6;   A,  10.  18.    9  miles  an  hour. 


19.    miles  an  hour.     4  miles  an  hour. 


14:  am         „ 
-a±  V^^  +  ^ 

20.  Length, ^— ^^^^' 

weight, /,  lbs.      Length,  8  feet; 

^    A  arn,        „  .,,.-,, 

—  a±\/ \-a^  weight,  4^  lbs. 

21.  <^VnTmVc  Vn(VnTVc)  ^^^^^^^. 

^Jn^  \/c  a-m 

Vc(Vn^:Vc)  ^^^^^^^     -^g  ^^yg .  j^  ^1  25 ;  B,  $1 , 
a  —  m 

22.  10  (—  5  ±  Vfn  +  25)  dollars. 


23.    10(5  ±  V25  —  m)  dollars. 

Exercise  120. 

1.  ±2,  -fcVlO;  1,-2;  2, -1, -1±  ^=3,  ^(1±a/^). 

2.  ~3;   ±  a,   ±5.  4.    16,  ifffi-;  8,  W- 

3.  ±4,  ±  i;  9,  (-41)1  5.    ±  8,  ±(-11)^;  9,  ^/i681. 


ELEMENTS  OF  ALGEBKA.  t 

6.    3»,  (-28)1;    (J)«,  1.  9     i     JL  .>_,.,     , 

tt-     4  a-               *     ^    ^ 
'■   '«.li   27.  ,        1 

8.  jm;«)u.         -1,(196)".  Venn  (j^ 

10.  ±^^/2^,  ±  \^^;  ±  vS,  ±  v'-S. 

Query.     Wliy,  iu  10,  the  ±  sign  ? 


Exercise  121. 

1.  2,  -  3,  -1  ^  3V5  .  9.    i(3  ±  V5),  i(9  ±  ^f^^\ 

2  5±  V37    5±  V7 

2.  -1, -1±2V15.  "'  3        '        3       ' 

3.  ±  4  V6;   18^,  5.  ^^'    -'  ^'        _ 

-  3  ±  V33 

4.  3,  -i,  4  C5  ±  V1329).      13.   1^ ,  -1,-2. 

5.  5,-6,  K-l±  V377).     14-    1,  2,  -5,  8. 

®-  '  ^'         3  16.    5,  - 1,  2  ±  V5. 

7.    1,-3±2V2.  -3±a/29    -3±  \/I^ 

—  ^'^'  4  '  4 

8. 4, 1,^^^.         18. 2^e^,  ^rs. 

10.   3,  —1. 
19.   ^-^^,  -1±  V2;  3,-5,  , -2  (-13  ±  V313). 

Exercise  122. 

1.  10,-2;   7,  14.  4.    14,  2.48;    V- 

2.  9,  -12.  5.   4,  -21;  5,  \. 

3.  i,  V;   8.  -I-  «•   ^'  4'  -^5    12,  4. 


84  ANSWERS  TO  THE 

7     _|_  9  V2;   3,  ^.      Query.     Is  ^  a  true  value  for  x  in  the 
second  equation  ?     Why  ? 

8.  0,  ±  V3.  12    i^    ^-   4    B4 

2m^/n  ^        ^ 

9.  2^T.  §;    ±     ^^  _^  ^    •  13.    ±  (^;    9,  V-  

10.  0,4/7;   ±Vl±h^2.       14.    1,  4;    9,  4,  ~  ^     ^    "  ^ ' 

11.  ±*-'(^')';  25,  -V-    16,   0,  2,-3;    ±  ^^  V3. 
_    7±Vl3    -lq=V^^     ,,    ,    o-^o./^ 


.u. 

2"       '             2            )    -^'^j   -^j  ^  -r- 

^     V    t  , 

Exercise  123. 

1. 

-8,-9;   W,  49. 

2. 

.  J-;;, 

mn 

m  —  n^ 

3. 

0,  ^;    a,  a^    15,  -3. 

2  b,  y'-a'. 


5.  a;2_4a.^21;  Gcc^  +  Sx^G;  a;2-2a;  =  15;  x'  =  -Z\ 

6.  a;2  +  5a:  =  0;   x''-Ux  =  —29',   x^  —  2  x  =  1. 

7.  ic'-^  —  (1  +  m^)  X  =  m  {1  —  rn^)  ;  mnx^  —  (m^  —  ?i^)  a; 

=  m  w  ;  (a  —  Z*)  ic^  —  4  tt  6  ic  =  (ct  —  ^)  (a  +  hf-. 

8.  ic^  +  2  m  a;    =    8  /^  —  m- ;     4  ic'^  —  4  a  ic    =    Z»  —  a""^; 

Vox^  —  '6\/'(ix^h  —  a. 

Exercise  124. 

1.  Imaginary ;  real  and  equal  ;  imaginary. 

2.  Eeal  and  rational,  and  different;  imaginary. 

3.  Real  and  rational,  and  different;  real  and  surds,  and 

different. 


KLEMKNTS  OF   ALGEBRA  85 

4.  Real  and  equal ;  imaginary. 

5.  Imaginary ;  real  and  different,  and  surds. 

6.  Real  and  equal;  real  and  rational,  and  different;  real 

and  rational,  and  different. 

7.  Real  and  rational,  and  different,  real  and  rational,  and 

different ;  real  and  surds,  and  different. 

8.  Real  and  rational,  and  different;  real  and  surds,  and 

different. 

9.  Im;i<^nnary;  real  and  rational,  and  different. 

Exercise  125. 

1.  ±8,  ±(-ll)i;   h  -i;   h  4". 

2.  1^,  T^;    -1,  -ih'  3.    I  450;    4,  f 

^"/l  ±  \/l3     ^/l  ±  V-  7       J  ±  Ks  ±  2  \/4  g  +  1 
V         2        '    ^  2         '  '       2 

5.  ±  3, —- ;   5,  -4, 

6.  ±  h  h  h  7.   4,  1,  3,  2. 

9.    ±  2  \/2,  ±  \  a/=^,    ±  i  V^185  ±  29  Vil. 

„    n    3±4V3     ,     ,  "•    4m»,  to';    -8,-4. 

X2.    0,  -^— ,    1,  4.  ^^     ^_  J 

13.   4,  1;   J,  y.  16.   0,  1,  3. 


86  ANSWERS  TO  THE 

17.  -a±-— ^=;    4,  3^3. 

18.  (i.)    a/^^  —  4  ^  a  surd  ;     (ii.)    ^^  _  4  ^  positive; 
(iii.)    ^2  _  4^  negative;    (iv.)    yl^-4j5  =  0; 

(v.)    J5  positive;  (vi.)    B  negative; 

(yii.)    B  =  --. 


j^.'±V4j^+a--|,^g 


Exercise  126. 


12/ =  5,  4.  ■    j2/  =  9,- 


2. 


r.x  =  S, -1,  J  a;  =  7, -5, 

l2/=i,-2.  ■1j,  =  5,-7. 

.^  =  6,-4,  11     \x  =  %-5, 

'•  ly  =  3,-7.  ly  =  5,-9. 

p-=  6,^^,-41,  p  =  -7.4,1, 

*-J3,  =  3i,-4.  1^  =  7.8,5. 

.    rx  =  18,  12J,  <x  =  ll, -8, 

=-iy  =  3,-2i.  "•1y  =  8,-ll 

^•iy  =  6,3.  "•1^  =  1. 

p  =  4,3,  ^^    |.  =  5,4, 

|y  =  3,  4.  l.y  =  4,  5. 

(x  =  8,17§,  ig     fa;  =  ll,-7, 

°-    l2,  =  6,-13i.  l3/  =  7.-ll. 
Query.     In  14,  has  x  and  y  two  values?     Why.' 


1. 


ELEMENTS  OF  ALGEBRA.  87 

Exercise  127. 

(x  =  ±5,±7y  |a:  =  3, -3,  2,-2, 

U=±7,  ±5.  •   -^2/  =  2, -2,3,-3. 


(X=  i, 


7,  4,  f  X  =  5,  —  5,  1,  — ,  1, 


7.  *  j  y  =  1,  - 1,  5,  —  o, 

p  =  3,2,  (x=±5,  ±3, 

^-   U  =  2,3.  ''•   |y=±3,±5. 

J  a:  =  5, -5,  3,-7,  j  x  =  10,  4. 

*•   1^  =  3,-7,5,-5.  •   1//-4,10. 

ra;  =  6,  4,  ^    ,3.=  ±  2,  ±3, 

'    (y  =  4,6.  •  |y=±  3,  ±2. 

p  =  6,-6,5,-5,  (0^  =  5,3, 

^-  |y  =  5,-5,6,-6.  ^^-   (y  =  3,5. 

Exercise  128. 

r  x  =  ±  2,  ±  1, 
•  ly=±l,  ±2. 

rx=±2, 
<x  = 


1. 

{" 

±3, 
±2. 

2. 

!;: 

±6, 
±  2. 

3. 

i;: 

±3. 

4. 

\l' 

:±2, 

:   ±3. 

5. 

■■±h 
■■±h 

±i. 

±3- 

7. 


±  7,  ±  >v/3,_ 
±2,  TSV3. 


(  ar  =  ±  3,  ±  36, 
•    -j  y  =:  ±  5,  :f  ¥• 


r  X  =  ±  8, 
1  y  =  =F  5, 


^^^x=±8,±3. 


88 


ANSWERS  TO  THE 


1.  r 


2. 


3. 


5.  r 

(y 

6.  f 

7.  f 


3,5, 

o,  3. 

6,-3, 
3,  -6. 

8,-7, 
7,-8. 

10,  12, 
12,  10. 


10. 


11. 


12. 


13. 


Exercise  129. 
14.]^ 

\y 

15. 


(2/  = 


y 


16.  < 


5, -3,l±iV-88, 
-3,  5,  1  T  iV--88. 

3,2, 
2,3. 

,   o  -13±V377 

4,  ^,  ■ -^^^ , 


{y  = 


±  7,  ±  5, 

±  5,  ±  7. 

a: 

2, 

17. 

3. 

y 

20, 

15. 
0,  5,  1, 

18.    . 

(0! 

0,1,-1. 

19.   . 

P 

±10, 

liS/ 

±4. 

20.   . 

^Vf 

21.   < 

a; 

T  1,  T  2  a/=3, 

2/ 

±  i,  ±  V- 

-1. 

0,4, 
0,5. 

22.    . 

[y 

0,-1, 
0,-2f. 

23.    . 

y 

y  =  2,  4, 


=  4,3, 


=  3,4, 


6 
-13:Fi 

/377 

6 

7±  V- 

295 

2 

7:f  V- 

295 

2, 

-2J, 

4, 

-4|. 

6, 

-2, 

2, 

-6. 

2, 

h 

2, 

16. 

5  IF  V15 
2 

5±  Vi5 


±iv/l8±2V-li 
i(l±V-15). 


1,2, 
3,1. 


ELEMENTS  OF  ALGEBRA.  89 


26 


27 


f  1  -\-ab±  V(«  +  1)  (fl^  -  1)  (b  +  1)  (6  -  1) 

^*'    j  «&  -  1  ±  V(a  +  1)  (a  -  1)  (b  -\-l)(b-  1) 

1^^"  a-b 

rx=±9,  ±3,  (:r  =  2,  i, 

-*^-  -iy=:±3,  ±9.  •   -1^  =  3,-24. 

(  X  =  ±  5,  ±  2,  ±  2  v^,  ±  6  V^, 

1 2/  =  ±  2,  ±  5,  ±  5  V-  1,  ±  2  V-H. 

(x  =  ±l,  ±3,  jx  =  3,  2, -3±  v^, 

(y=  ±3,±1.  •   1y  =  2,3,-3T  V3. 

„     (  «  =  ±  2,  ±  3,  ^  a:  =  7,  1,  4  ±  2  V7, 

''"•   1y=±3,±2.  •   ■iy:=:l,7,4:f  2V7. 

30    -^ 

■[y  =  °a^V3),j(i.-L). 

rx  =  2,4,3q:v'21,  (  a:  =  5,  4, 

^^-   |y  =  4,2,  SiVai.  ■<y  =  4,5. 

32.  r=o'"!'        •«>r==*^!'*l' 

(y  =  2,-4.  <y  =  ±4,  ±9. 

^^Tft'  ■1y  =  l,9. 


33. 


35. 


„=±?1±*!.  „    5a;  =  62'S,l, 

^       *   a-4  "•    (y  =  1,625. 

x=  ±4,  itV^^eB,  -va6±V34lo), 


90  ANSWERS  TO  THE 


^x=±  (a-h),  ^x  =  h-h 


^      =1,5.  50. 


2/=  ±^' 


ly  =  ^, 

46.  .^2/ -^,5,-,, 

47.  j^  =  ^^'  52.    9  and  7. 

Query.     In  47,  how  many  values  has  x  and  y  ?     Why  ? 


63.    ±  }^V^((^  +  ^)  and   ±  ^  V2  (tt  -  6).     11  and  7. 

54.  36. 

a  +  ^-3a^  T  2  V2  (a""  +  bj 

55.  __zi_I Z_ r — V — 2_y    ^^^ 

Z 

a  +  ^-  3  ^2  +  2  V2  (a^  +  Z>)       Q      ^  1         1        1  Q 
— - — ^^ .     3  and  1,  or  1  and  3- 

Z 

56.  —  a  and  —  2  a,  or  2  a  and  a. 

57.  5   and   2,    or   —  2   and   —  5,    or   ^  (3  ±  V—  67)   and 

i  (-  3  +  V=^67). 

58.  6.4,  or  4.6.  59.   3,  15,  and  20. 


2  c"  •       ,  —  a  ±  V4  c^  +  a^ 

61.   =:^=  and .     8  and  4,  or 

.       —  a  ±  \/4:  (§  +  a^  2 

—  4  and -8. 

62.  f ,  or  -^f.  60.    4  and  13. 

63.  Time,  7  years,  or  6  years ;    rate,  .06,  or  .07. 

64.  Principal,  $10400;  rate,  .05. 


ELEMENTS  OF  ALGEBRA.  91 

Exercise  130. 

^     (x  =11,8,5,2;  x  =  2,7,12,17,  ....;  a;  =  8, 16, 24, 32, .  .. 
^'   1^  =  1,3,5,7;  ^  =  7,21,35,49,....;  i/ =5,8, 11, 14,.... 

^  a;  =  42,  31,  20,  9;  x  =  215,  202,  189,  176,  . . . . ,  7. 
^    -^y  =  4,  9,  14,  19;  y  =  5,  14,  23,  32,  ....,  149. 

X  =  8,  25,  42,  59,  ,.. .;   x  =  7,  16,  25,  34, ....; 

x  =  4,  17,  30,  43,  .... 
y  =  7,  22,  37,  52,  ....;  y  =  10,  23,  36,  49,  ....; 

y  =  2,  11,20,29,.... 

^x  =  6;x  =  9;x  =  0.  ^x  =  10, 

|y  =  3;  2/  =  3;  y  =  15.  '   \y  =    5. 

IX  =  3;  x  =  59;  x  =  476.  ^  x  =    4, 

ly  =  '2;y=    1;3,=    19.  "   ^  =  24. 


6. 

r  X  =  11 ;  X  =  37. 
jy=18;2/=13. 

Exercise  131 

11. 

2. 

59. 

6. 

\  and  I 

3. 

13  and  1,  or  4  and  8. 

7. 

131. 

4. 

72  and  70. 

9. 

Nine  ways. 

5. 

(  5  foot  rod ;  19, 12,  and  5. 

11. 

No. 

foot  rod ;  4, 9,  and  14. 

10.   Two  ways  in  each.     39  and  6. 

12.    A  gives  B  thirteen  50-cent  pieces,  and  B  gives  A  forty- 
five  3^ cent  pieces. 

r  Horses,  1,3,  5,  7,.... 
(Sheep,  8,23,38,53,.... 


92  •        ANSWERS  TO  THE 

rl6,  15,  14,  13,  ....,at^25. 

14.  J    2,    5,    8,11,....,  at  $15. 

157,  55,  53,  51,  ....,  at  $10.     Sixteen  ways. 

r   6,  13,  20,  27,  34,  at  30  cents. 

15.  \  45,  35,  25,  15,    5,  at  45  cents. 

1 24,  27,  30,  33,  36,  at  80  cents.     Five  ways. 

(-74,  73,  72,  ....,  at  20  cents.     . 

16.  ^4,    8,  12,  .... ,  at  35  cents. 

I  72,  69,  66,  ....,  at  40  cents.     Twenty-four  ways. 

r23,  16,    9,    2,  at  $1.50. 

17.  \  13,  16,  19,  22,  at  $1.90. 

l   4,    8,  12,  16,  at  $1.20.     Four  ways. 

18.  Four  ways.  21.   269. 

19.  4  at  $19,  8  at  $7,  and  8  at  $6. 

22.  The  first  has  63  or  23,  the  second  37  or  77. 

21,  23,  25,  27,  poles    7  feet  long. 

23.  ^  18,  13,    8,    3,  poles  10  feet  long. 
1,    4,    7,  10,  poles  12  feet  long.       Four  ways. 


Exercise  132. 

1. 

a;>i|;  a^  <  4^- 

12    1"^^^' 

2. 
3. 

x>-^\  X  <b- 
x>l. 

-1. 

13.    f>^^ 

9. 

x  =  6. 

ly  <  a-b. 

10. 

X  =  4:. 

15.   76  or  77. 

11. 

x  =  5. 

16.    60  cents. 

ELEMENTS  OF  ALGEBRA.  93 


Exercise  133. 
8.i^2.  a  -\-  b        2ab      m        n  ^    1      .1 

5.  '^±y>t^.    3(l  +  a'^  +  a^)>(l  +  a  +  a«)^ 
X  —  y      X*  —  y^  ^  ' 

6.  If«>/;,  a»  +  26«>3(^6^    :^  >  :^;     V2  4-V7 

_  v6        v3 

>  V3  +  V5. 

Exercise  134. 

5.  4x^-3x4-1  >  6.  18    j2^<^' 

6.  4-2a-'+;i2>3^^^  '    <  x  >  0. 

7.  3  +  A>a^  +  3:c«.  19-   j^Jj^^^' 

8.  3(a:-y)^>3(ci4-<^r;    1^  (x^  +  y)  >  a;«  -  y» 

10.  lln'»>a2  +  8  6.  '    \x  >V- 

11.  w^  <  54;    „  c  <  7i«-^2^  ^  a;  <  _  ^, 


12.  -8>-27;  15>8;5>3. 

13.  -3>-4;  4m2  +  1>7i.  24.    2. 

14.  w  >  w ;    2  >  —  4.  25.    6,  or  7. 

15.  2aj>m-7i;  -««>/.  26.    16,  or  17. 

16.  x<  2.9;  X  <3i.  27.    19. 

17.  jc<|;    ar<i.  37.    13. 

22.  x>  5,  ar<^;   a;>-i,  x<-§. 

23.  a;  >  §,  X  <  i ;    x  >  7,  x  >  -  3. 
28.  2,  1,  0,  -1,  -2,  -3,  -4;   -8. 


Vm  ^  Vn 


miles.    126  miles. 


94  ANSWERS  TO  THE 


Exercise  135. 

1. 

-49. 

5.    SO  a  — 79  b. 

9. 

12. 

2. 

161,  245. 

6.    -1,0. 

10. 

19. 

3. 

16,9. 

7.    i. 

11. 

-95. 

4. 

98,  243.6. 

8.    103. 
Exercise  136. 

12. 

2,  21,  and  2^. 

1. 

779. 

4.   If  (9-.). 

7. 

76  a  4-  57  h. 

2. 

-  5569|. 

3.--^ 

8. 

8^. 

3. 

a^{4:-a). 

6.    a7i(?z+l) 

9. 

13. 

10.  (1)   71  =  16,  d  =  —  l.     (2)   7i  =  7,  d  =  2  a. 

11.  5^,  6^,  7t,  etc.         13.    7500.         14.   -5,-1,3,7,11. 

Exercise  137. 

1-   -^H'-6A,  ••••, -^T-V    5.    x''+l-x,x^+2-2x,....,x. 

7n    -4-   7i 

2.  6.4,  5.6,  ....,-5.6.  6.       ,  ,. 

3.  4  ?w-  —  5  ?z,  3  m  —  4  w,  — ,  6  7i  —  5  m. 

4.  _2l,-3§, -4f, -5|.        7.   -101,-7^. 


1.   48;  384. 


8.  ^^W,  Hi  W,  ••.. 


Exercise  138. 

3-    —  ^t;   —  2T^T- 

4.    128;  1. 

6.    x^»'- 

,-288.                  32^ 
^'    243' 

«  (§)'• 

ELEMENTS  OF  ALGEBUA.  95 


Exercise  139. 

1.   2|f 

4.    V(3+V3). 

7.    .'5. 

520. 

2.  mh' 

5.    H3'"-l). 

8.    2. 

3.  mh 

6.    §(1-4'"). 

9.   6. 

5,  30,  180. 

Exercise  140. 

1.  42;   aH^',   J  V6;   H-  5.   40,  16,  6§. 

2.  ik;  6x2  — ox  — 6.  g    _7^  ^,  _|,  1^  _^,  ^^. 

3.  20,  80.  8.   Arithmetical,  f . 

4.  i,  i.  9.   2  and  8. 
7.  6,  18,  54,  162,  486,  1458,  4374. 

Exercise  141. 

1.  -4.  11.   1,9. 

2.  ^.  12.   20,5. 

3.  —  ij't,  —  5V'  —  ^>  — '  iV«  ^^f)  — 

4.  4,  2,$,  I,  f  ....  13.   10,12,15. 

5.  If  14.   i,i,i,  01-^,1, -J. 

6     ''4,6.  ^_,_^  2a6 

15.    —^,  Vab, 


8.  f ,  5'  A,  h  W,  A.  !«•   2550  yards. 

9.  6,  12.  17.   3,  6,  12,  24. 
10.   6,  2. 


96    ANSWERS  TO  THE  ELEMENTS  OF  ALGEBRA. 


Exercise 

142. 

1. 

54  6  :  a. 

10. 

-13. 

2. 

9:7. 

11. 

2,  or  3. 

3. 

10  :  9. 

12. 

7:  2. 

4. 

(l-y)(l  +  x):l+x\ 

13. 

an  —  hm 

5. 

14. 

m  —  n 
h  in  —  an 

6. 

5a:4;  Ay  :  6x\  Sy  : 

X. 

m 

7. 

28 ic  :  By,  n  —  1  :  2a. 

15. 

4:7. 

8. 

7  :  8,  31 :  36,  41 :  48,  5 

:6. 

16. 

"^ah. 

9.    a"  —  h^-.a"-  -\-h'-  >a  —  h\a^-h. 

Exercise  143. 

3.  4;  2;  t33-V210;  12;  ^2^2^    29.    h  '.  a. 

4.  ^2  _  ^2 .  300  ^8  j^^  g^     ^^ 

5-    M;  3r\;  .8;  4^;  1^.  3-,^    4  .  1^ 


a?" 


6.  3";   (a  +  ^)2;  ^f--,-.-  32.  17:7. 

7.  15;  i;  2^3^;  6^-JZ».  ^^-  '"^  *  ^• 

8.  5/;2;  1.  34.  5:  4. 

21.    a  =  c'  c?3,  ^  =  c^  (^i  35.  V&xy. 

23.    a.=  t^^a^^;a:-3,or-l.  37.  ^  =  ^' ■ 

^^  '  b       n^ 


24. 


( a;  z!r  i  4,  i-  6,  ^^a  +  bV         „  . 

^  ^         /  39.    £c  =  ^;    — ^7  );  ir  =  3,or— 1. 

(y  =  ±6,  ±4.  W^y 


27.   A  invested  $3000  ;  B,  $3500. 

38.    Bate  of  slow  train  ;  rate  of  fast  train  : :  1  :  2. 


or  THE     ^  ^ 

f    UMIVERSITY  I 

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